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Constructions – CBSE NCERT Study Resources

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9th

9th - Mathematics

Constructions

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Chapter Overview: Constructions

This chapter introduces students to the fundamental geometric constructions using a compass and straightedge. It covers the basics of constructing angles, triangles, and perpendicular bisectors, as well as more advanced constructions like angle bisectors and specific types of triangles based on given conditions.

Geometric Construction: The process of drawing geometric figures accurately using only a compass and an unmarked straightedge.

Key Topics Covered

  • Construction of angles (e.g., 60°, 90°, 120°)
  • Construction of perpendicular bisectors of line segments
  • Construction of angle bisectors
  • Construction of triangles given base, angles, or sides

Detailed Concepts

1. Basic Constructions

Students learn to construct simple geometric shapes and angles using minimal tools. Examples include:

  • Constructing a 60° angle using a compass
  • Drawing a perpendicular bisector of a given line segment

Perpendicular Bisector: A line that divides a given line segment into two equal parts at a right angle (90°).

2. Construction of Triangles

This section focuses on constructing triangles under specific conditions, such as:

  • Given base, base angle, and sum of other two sides (SSA condition)
  • Given base, base angle, and difference of other two sides
  • Given perimeter and two base angles

3. Advanced Constructions

Students explore more complex constructions, including:

  • Constructing an angle bisector
  • Constructing a triangle with given side lengths (SSS condition)

Angle Bisector: A line or ray that divides an angle into two equal parts.

Practical Applications

Understanding geometric constructions helps in solving real-world problems, such as designing structures, creating accurate diagrams, and understanding symmetry in shapes.

Summary

The chapter equips students with the skills to perform precise geometric constructions, laying the foundation for more advanced topics in geometry. Mastery of these techniques is essential for solving problems involving angles, triangles, and other geometric figures.

All Question Types with Solutions – CBSE Exam Pattern

Explore a complete set of CBSE-style questions with detailed solutions, categorized by marks and question types. Ideal for exam preparation, revision and practice.

Very Short Answer (1 Mark) – with Solutions (CBSE Pattern)

These are 1-mark questions requiring direct, concise answers. Ideal for quick recall and concept clarity.

Question 1:
What is the purpose of constructing parallel lines?
Answer:
To maintain equal distance throughout.
Question 2:
In constructing a triangle, how many sides and angles are needed?
Answer:
  • 3 sides
  • 2 angles
Question 3:
What is the first step in constructing a perpendicular bisector of a line segment?
Answer:
Draw arcs from both ends of the segment.
Question 4:
How many arcs are needed to construct a 60° angle using compass and ruler?
Answer:
2 arcs
Question 5:
What is the angle between the bisector and the original line?
Answer:
90°
Question 6:
Name the tool used to measure angles in constructions.
Answer:
Protractor
Question 7:
What is constructed when two perpendicular bisectors of a triangle intersect?
Answer:
Circumcenter
Question 8:
How do we verify a constructed angle bisector is accurate?
Answer:
Check if it divides the angle equally.
Question 9:
What is the radius used to draw arcs in constructing a 30° angle?
Answer:
Same as the line segment length.
Question 10:
Which geometrical shape is used to construct a 45° angle?
Answer:
Square
Question 11:
How do you construct an angle of 60° using a compass and ruler?
Answer:

Draw a baseline OA.
With O as center, draw an arc intersecting OA at P.
With P as center, draw another arc intersecting the first arc at Q.
Join OQ, forming a 60° angle.

Question 12:
What is the purpose of constructing an angle bisector?
Answer:

An angle bisector divides a given angle into two equal parts.
It ensures symmetry and is used in geometric proofs, designs, and real-world applications like architecture.

Question 13:
List the tools required for geometric constructions.
Answer:
  • Compass for drawing arcs and circles
  • Ruler for measuring and drawing straight lines
  • Pencil for marking points
  • Eraser for corrections
Question 14:
How do you verify if a constructed angle is accurate?
Answer:

Use a protractor to measure the angle.
Alternatively, compare it with a standard angle using tracing paper or geometric properties.

Question 15:
What is the significance of keeping the compass radius greater than half the line segment length while constructing a perpendicular bisector?
Answer:

This ensures the arcs intersect at two distinct points.
If the radius is too small, the arcs won’t intersect, making the construction impossible.

Question 16:
Explain how to construct a 30° angle using a compass and ruler.
Answer:

First, construct a 60° angle.
Then, draw its angle bisector to split it into two 30° angles.

Question 17:
Why must the compass needle be fixed at the endpoint while drawing arcs in angle construction?
Answer:

Fixing the compass at the endpoint ensures the arc is drawn with the correct radius.
This guarantees the accuracy of the constructed angle or bisector.

Question 18:
What is the difference between bisecting a line segment and bisecting an angle?
Answer:
  • Line bisector divides the segment into two equal lengths at 90°.
  • Angle bisector splits the angle into two equal angles without any fixed perpendicularity.
Question 19:
Can you construct a 45° angle without constructing a 90° angle first? Justify your answer.
Answer:

No, because a 45° angle is half of a 90° angle.
You must first construct a right angle and then bisect it to get 45°.

Very Short Answer (2 Marks) – with Solutions (CBSE Pattern)

These 2-mark questions test key concepts in a brief format. Answers are expected to be accurate and slightly descriptive.

Question 1:
Construct an angle of 90° using a compass and ruler. Write the steps involved.
Answer:

To construct a 90° angle:
1. Draw a straight line AB using a ruler.
2. Place the compass at point A and draw an arc that intersects AB at C.
3. Without changing the compass width, place it at C and draw another arc intersecting the first arc at D.
4. Draw a line from A through D. The angle formed is 90°.

Question 2:
What is the purpose of an angle bisector in constructions?
Answer:

An angle bisector divides a given angle into two equal parts. It ensures precision in constructions where equal angles are required, such as in triangles or polygons.

Question 3:
Construct a 60° angle using a compass and ruler. List the steps.
Answer:

To construct a 60° angle:
1. Draw a line segment PQ.
2. Place the compass at P and draw an arc intersecting PQ at R.
3. Without changing the compass width, place it at R and draw another arc intersecting the first arc at S.
4. Draw a line from P through S. The angle formed is 60°.

Question 4:
Explain how to construct the perpendicular bisector of a line segment.
Answer:

To construct a perpendicular bisector:
1. Draw a line segment AB.
2. With A as the center, draw arcs above and below AB using a compass.
3. Repeat step 2 with B as the center, ensuring the arcs intersect.
4. Join the intersection points to form the perpendicular bisector.

Question 5:
Why is it important to keep the compass width unchanged while constructing angles?
Answer:

Keeping the compass width unchanged ensures consistency in the radius of arcs drawn, which is crucial for accurate angle or shape constructions. Changing the width may lead to incorrect measurements.

Question 6:
Construct an angle of 120° using a compass and ruler. Describe the steps.
Answer:

To construct a 120° angle:
1. Draw a line segment XY.
2. Place the compass at X and draw an arc intersecting XY at Z.
3. Without changing the compass width, place it at Z and draw another arc intersecting the first arc at W.
4. From W, draw another arc to intersect the previous arc at V.
5. Draw a line from X through V. The angle formed is 120°.

Question 7:
What is the significance of the point of intersection in constructing a perpendicular bisector?
Answer:

The point of intersection of arcs is where the perpendicular bisector passes through. It ensures the line is equidistant from both ends of the segment, making it a true bisector.

Question 8:
Construct a 30° angle using a compass and ruler. Outline the steps.
Answer:

To construct a 30° angle:
1. First, construct a 60° angle using standard steps.
2. Bisect the 60° angle using an angle bisector.
3. The resulting angle will be 30°.

Question 9:
How does constructing an equilateral triangle help in angle construction?
Answer:

An equilateral triangle has all angles equal to 60°. By constructing it, we can derive other angles like 30° (by bisecting) or 120° (by combining two 60° angles).

Question 10:
Describe the steps to construct a 45° angle using a compass and ruler.
Answer:

To construct a 45° angle:
1. First, construct a 90° angle.
2. Bisect the 90° angle using an angle bisector.
3. The resulting angle will be 45°.

Question 11:
What is the purpose of constructing the perpendicular bisector of a line segment?
Answer:

The perpendicular bisector divides a line segment into two equal parts at 90°.
It is used to:

  • Find the midpoint of a line.
  • Construct geometric shapes like triangles and quadrilaterals accurately.
  • Ensure symmetry in designs.

Question 12:
Explain how to construct an equilateral triangle with a given side length.
Answer:

To construct an equilateral triangle:
1. Draw a line segment AB of the given length.
2. With A as the center, draw an arc of radius AB.
3. With B as the center, draw another arc of the same radius intersecting the first arc at C.
4. Join A to C and B to C to complete the triangle.

Question 13:
What is the significance of the angle bisector in constructions?
Answer:

The angle bisector divides an angle into two equal parts.
It is used to:

  • Ensure accuracy in geometric designs.
  • Construct angles of specific measures (e.g., 45° from 90°).
  • Create symmetrical shapes and patterns.

Question 14:
Construct a 45° angle using a compass and ruler. Describe the steps.
Answer:

Steps to construct a 45° angle:
1. First, construct a 90° angle using perpendicular lines.
2. Draw the angle bisector of the 90° angle.
3. The bisector divides the angle into two equal 45° angles.

Question 15:
How do you construct a rhombus with a given side length and one angle?
Answer:

To construct a rhombus:
1. Draw a side AB of the given length.
2. At point A, construct the given angle using a protractor or compass.
3. Mark the next vertex D at the same side length.
4. Repeat the angle and side length to mark C and complete the rhombus.

Question 16:
Why is it important to keep the compass width unchanged during certain constructions?
Answer:

Keeping the compass width unchanged ensures:

  • Equal radii for arcs, which is critical for accuracy.
  • Consistency in geometric shapes (e.g., circles, angles).
  • Correct alignment of points in constructions like perpendicular bisectors or triangles.

Question 17:
What precautions should be taken while constructing geometric shapes?
Answer:

Precautions include:

  • Using sharp pencils for precise lines.
  • Ensuring compass needles are fixed tightly.
  • Double-checking measurements and angles.
  • Avoiding smudges to maintain clarity.

Short Answer (3 Marks) – with Solutions (CBSE Pattern)

These 3-mark questions require brief explanations and help assess understanding and application of concepts.

Question 1:
Construct a triangle ABC where AB = 5 cm, BC = 6 cm, and AC = 7 cm. Write the steps of construction.
Answer:

To construct triangle ABC with given sides:

  • Draw a base line segment BC = 6 cm.
  • With B as center, draw an arc of radius 5 cm (AB).
  • With C as center, draw an arc of radius 7 cm (AC).
  • The intersection point of the two arcs is A.
  • Join A to B and C to complete the triangle.

This method ensures the triangle follows the given side lengths accurately.

Question 2:
Construct an angle of 75° using a compass and ruler. Explain the steps.
Answer:

To construct a 75° angle:

  • First, draw a baseline OA.
  • With O as center, draw a 60° angle using standard construction.
  • Bisect the 60° angle to get 30°.
  • Now, construct a 90° angle adjacent to the 60° angle.
  • Bisect the 30° angle between the 60° and 90° to get 15°.
  • Add 60° + 15° = 75° to get the required angle.

This combines angle addition and bisection techniques.

Question 3:
Construct a perpendicular bisector of a line segment XY = 8 cm. Describe the steps.
Answer:

To construct the perpendicular bisector of XY = 8 cm:

  • Draw XY = 8 cm.
  • With X as center, draw arcs above and below the line using a radius > 4 cm.
  • Repeat with Y as center, same radius.
  • Mark the intersection points of the arcs as P and Q.
  • Join P and Q to form the perpendicular bisector.

The bisector will divide XY into two equal parts at 90°.

Question 4:
Construct a triangle PQR where PQ = 4 cm, ∠Q = 45°, and QR = 5 cm. List the steps.
Answer:

To construct triangle PQR:

  • Draw base QR = 5 cm.
  • At point Q, construct a 45° angle using a protractor or compass.
  • From the angle line, mark PQ = 4 cm.
  • Join P to R to complete the triangle.

This ensures the triangle meets the given side and angle conditions.

Question 5:
Construct a line segment AB = 6.5 cm and divide it into four equal parts using a compass and ruler. Explain the method.
Answer:

To divide AB = 6.5 cm into four equal parts:

  • Draw AB = 6.5 cm.
  • Construct the perpendicular bisector of AB to find the midpoint M.
  • Now, bisect AM and MB separately to get points P and Q.
  • This divides AB into four equal segments of 1.625 cm each.

Repeated bisection ensures equal division without measurement errors.

Question 6:
Construct an angle of 75° using a ruler and compass. Justify your construction steps.
Answer:

To construct a 75° angle, follow these steps:

1. Draw a baseline AB using a ruler.
2. With A as the center, draw an arc intersecting AB at C.
3. With C as the center, draw another arc to intersect the first arc at D, creating a 60° angle.
4. Bisect the 60° angle to get a 30° angle.
5. From the 30° angle, construct an adjacent 45° angle (by bisecting a 90° angle).
6. The sum of 30° and 45° gives 75°.

Justification: The construction combines basic angle measures (60°, 90°) and bisection to achieve the desired angle.

Question 7:
Construct a triangle ABC where AB = 5 cm, BC = 6 cm, and ∠B = 60°. Explain the steps.
Answer:

Steps to construct triangle ABC:

1. Draw the base BC = 6 cm using a ruler.
2. At point B, construct a 60° angle using a protractor or compass.
3. From B, measure 5 cm along the new line to mark point A.
4. Join A to C to complete the triangle.

Verification: Measure the sides and angle to ensure accuracy. The triangle should satisfy the given conditions.

Question 8:
Construct a perpendicular bisector of a line segment XY = 8 cm. Describe the steps and its significance.
Answer:

Steps to construct the perpendicular bisector:

1. Draw XY = 8 cm using a ruler.
2. With X and Y as centers, draw arcs of equal radius (greater than half of XY) intersecting above and below XY.
3. Join the intersection points to form the perpendicular bisector.

Significance: The perpendicular bisector divides XY into two equal parts and forms a 90° angle. It is used in constructing equidistant points and geometric shapes.

Question 9:
Construct a right-angled triangle PQR where ∠Q = 90°, PQ = 4 cm, and QR = 3 cm. Verify your construction.
Answer:

Steps to construct triangle PQR:

1. Draw PQ = 4 cm horizontally.
2. At point Q, construct a 90° angle using a protractor or compass.
3. Measure QR = 3 cm vertically from Q.
4. Join P to R to complete the triangle.

Verification: Use the Pythagoras theorem to check if PR² = PQ² + QR² (i.e., PR = 5 cm).

Question 10:
Construct an equilateral triangle with each side measuring 5 cm. Explain why all angles are 60°.
Answer:

Steps to construct the equilateral triangle:

1. Draw a baseline AB = 5 cm.
2. With A and B as centers, draw arcs of radius 5 cm intersecting at C.
3. Join A to C and B to C.

Explanation: In an equilateral triangle, all sides are equal, so by the SSS congruence rule, all angles must also be equal. Since the sum of angles is 180°, each angle is 60°.

Long Answer (5 Marks) – with Solutions (CBSE Pattern)

These 5-mark questions are descriptive and require detailed, structured answers with proper explanation and examples.

Question 1:
Construct a triangle ABC where AB = 5 cm, BC = 6 cm, and ∠B = 60°. Explain the steps with proper justification.
Answer:
Introduction

We studied constructing triangles using SAS criteria. Here, we have two sides and an included angle.


Argument 1
  • Draw base BC = 6 cm.
  • At point B, construct ∠B = 60° using a protractor.

Argument 2
  • Mark point A on the new arm such that AB = 5 cm.
  • Join AC to complete ΔABC.

Conclusion

Our textbook shows similar problems. This method ensures accuracy in construction.

Question 2:
Construct a perpendicular bisector of a line segment XY = 8 cm. Justify each step.
Answer:
Introduction

We learned that a perpendicular bisector divides a line segment into two equal parts at 90°.


Argument 1
  • Draw XY = 8 cm.
  • With X and Y as centers, draw arcs of radius >4 cm intersecting above and below XY.

Argument 2
  • Join the intersection points to form the perpendicular bisector.
  • Verify by measuring angles and lengths.

Conclusion

This construction is useful in real-life designs like dividing land equally.

Question 3:
Construct an angle of 30° using a compass and ruler. Explain the steps.
Answer:
Introduction

We can construct a 30° angle by bisecting a 60° angle, which we studied in NCERT.


Argument 1
  • First, construct a 60° angle using an equilateral triangle method.
  • Draw an arc from the vertex to intersect the arms.

Argument 2
  • From these points, draw two intersecting arcs inside the angle.
  • Join the vertex to this intersection to get 30°.

Conclusion

This method is precise and used in architectural drawings.

Question 4:
Construct a triangle ABC where AB = 5 cm, BC = 6 cm, and ∠B = 60°. Justify each step with a brief explanation.
Answer:
Introduction

We studied how to construct triangles using a ruler and compass. Here, we are given two sides and an included angle.


Argument 1
  • Draw base BC = 6 cm using a ruler.
  • At point B, construct ∠B = 60° using a protractor.

Argument 2
  • From B, mark AB = 5 cm on the angle line.
  • Join A to C to complete ΔABC.

Conclusion

Our textbook shows similar constructions. This method ensures accuracy when two sides and an included angle are given.

Question 5:
Construct a perpendicular bisector of a line segment PQ = 8 cm. Explain the steps and its real-life application.
Answer:
Introduction

We learned that a perpendicular bisector divides a line segment into two equal parts at 90°.


Argument 1
  • Draw PQ = 8 cm and mark its midpoint M.
  • Using compass, draw arcs above and below PQ from P and Q.

Argument 2
  • Join the intersection points of arcs to form the bisector.
  • In real life, this is used in architecture to ensure symmetry.

Conclusion

Our NCERT examples demonstrate this clearly. The bisector ensures equal division and right angles.

Question 6:
Construct an angle of 45° using a compass and ruler. Verify your construction by measuring it.
Answer:
Introduction

We studied angle construction using bisectors. Here, we create 45° by bisecting 90°.


Argument 1
  • First, construct 90° by drawing perpendicular lines.
  • Bisect this angle using arcs from both arms.

Argument 2
  • Join the vertex to the intersection point of arcs.
  • Measure with a protractor to confirm 45°.

Conclusion

Our textbook shows this method. It is useful in designing right-angled structures like ramps.

Question 7:
Construct a triangle ABC where AB = 5 cm, BC = 6 cm, and ∠B = 60°. Justify your steps and mention one real-life application of such constructions.
Answer:
Introduction

We studied constructing triangles using a ruler and compass. Here, we need to draw ∆ABC with given sides and angle.


Argument 1
  • Draw base BC = 6 cm.
  • At B, construct ∠B = 60° using a protractor.
  • Mark A on the arm of ∠B such that AB = 5 cm.

Argument 2

Join AC to complete the triangle. Our textbook shows similar steps for constructing triangles with SAS criteria.


Conclusion

This method is used in designing triangular frames in architecture.

Question 8:
Construct a perpendicular bisector of a line segment XY of length 8 cm. Explain the steps and its significance in real-life scenarios.
Answer:
Introduction

We learned to draw perpendicular bisectors using compass and ruler. Here, XY is 8 cm.


Argument 1
  • Draw XY = 8 cm.
  • With X and Y as centers, draw arcs of radius > 4 cm intersecting above and below XY.

Argument 2

Join the intersection points to get the perpendicular bisector. NCERT examples demonstrate this for dividing lines equally.


Conclusion

This is useful in constructing symmetrical shapes like bridges.

Question 9:
Construct an angle of 90° using a ruler and compass. Describe the steps and justify how this method ensures accuracy.
Answer:
Introduction

Our textbook shows constructing right angles without a protractor. We use compass and ruler.


Argument 1
  • Draw a line segment AB.
  • With A as center, draw an arc cutting AB at P.
  • With P as center, draw another arc intersecting the first at Q.

Argument 2

Join AQ and extend to form 90°. NCERT confirms this method ensures precision.


Conclusion

This is applied in carpentry for right-angled joints.

Question 10:
Construct a triangle ABC where AB = 5 cm, BC = 6 cm, and ∠B = 60°. Justify your steps with geometric principles.
Answer:
Introduction

We studied constructing triangles using SAS criteria. Here, two sides and the included angle are given.


Argument 1
  • Draw base BC = 6 cm.
  • At B, construct ∠B = 60° using a protractor.

Argument 2
  • Mark point A on the new arm such that AB = 5 cm.
  • Join AC to complete ΔABC.

Conclusion

Our textbook shows this method ensures accuracy. The triangle satisfies the given conditions.

Question 11:
Construct a perpendicular bisector of a line segment XY = 8 cm. Explain how this is useful in real-life applications.
Answer:
Introduction

We learned that a perpendicular bisector divides a segment into two equal parts at 90°.


Argument 1
  • Draw XY = 8 cm. With X and Y as centers, draw arcs of radius >4 cm intersecting above and below XY.

Argument 2
  • Join the intersection points to form the bisector.
  • It helps in construction (e.g., dividing land equally).

Conclusion

Our textbook shows this ensures symmetry. It is used in architecture and design.

Question 12:
Construct an angle of 45° using a compass and ruler. Relate this to NCERT examples of bisecting angles.
Answer:
Introduction

We studied angle bisection to create smaller angles. Here, we derive 45° from 90°.


Argument 1
  • First, construct 90° by drawing perpendicular lines.

Argument 2
  • Bisect 90° using arcs from the vertex.
  • Our textbook shows this method in Chapter 11.

Conclusion

This technique is precise and used in designing tools like set squares.

Question 13:
Construct an angle of 45° using a compass and ruler. Explain the steps with reasoning.
Answer:
Introduction

We studied angle constructions in class. Here, we create a 45° angle using basic tools.


Argument 1
  • First, construct a 90° angle by drawing a perpendicular bisector.
  • Bisect this angle to get two 45° angles.

Argument 2
  • Use arcs of equal radius from the vertex to mark points on both arms.
  • Draw a line from the vertex through the intersection of these arcs.

Conclusion

Our textbook shows this method. It is used in architectural designs for precise angles.

Question 14:
Construct a triangle ABC in which BC = 7 cm, ∠B = 45°, and AB + AC = 13 cm. Write the steps of construction and justify your construction.
Answer:

To construct the triangle ABC with the given conditions, follow these steps:


Step 1: Draw the base BC = 7 cm using a ruler.
Step 2: At point B, construct an angle of 45° using a protractor and mark it as ∠B.
Step 3: From point B, measure a length of 13 cm (since AB + AC = 13 cm) along the angle line and mark it as point D.
Step 4: Join points D and C to form the line segment DC.
Step 5: Construct the perpendicular bisector of DC. The point where this bisector intersects BD will be point A.
Step 6: Join A to C to complete the triangle ABC.

Justification: The perpendicular bisector ensures that AD = AC, so AB + AC = AB + AD = BD = 13 cm, which matches the given condition. The angle at B is correctly constructed as 45°.

Question 15:
Construct a quadrilateral ABCD where AB = 5 cm, BC = 6 cm, CD = 4 cm, AD = 5.5 cm, and diagonal AC = 7 cm. Write the steps of construction and explain why this quadrilateral is unique.
Answer:

To construct quadrilateral ABCD with the given measurements, follow these steps:


Step 1: Draw the base AB = 5 cm using a ruler.
Step 2: With A as the center and radius 7 cm, draw an arc for the diagonal AC.
Step 3: With B as the center and radius 6 cm, draw another arc intersecting the first arc at point C.
Step 4: Join A to C and B to C to form triangle ABC.
Step 5: With A as the center and radius 5.5 cm, draw an arc for side AD.
Step 6: With C as the center and radius 4 cm, draw another arc intersecting the previous arc at point D.
Step 7: Join A to D and C to D to complete the quadrilateral ABCD.

Uniqueness Explanation: The quadrilateral is unique because all sides and one diagonal are fixed. The construction ensures that only one possible quadrilateral can satisfy all the given measurements, as the positions of points C and D are determined unambiguously by the intersecting arcs.

Question 16:
Construct a triangle ABC where AB = 5 cm, BC = 6 cm, and AC = 7 cm. Write the steps of construction clearly and justify each step.
Answer:

To construct a triangle ABC with the given sides, follow these steps:


Step 1: Draw a base line segment AB = 5 cm using a ruler.
Step 2: With A as the center, draw an arc of radius 7 cm (since AC = 7 cm) using a compass.
Step 3: With B as the center, draw another arc of radius 6 cm (since BC = 6 cm) using a compass.
Step 4: The point where the two arcs intersect is C. Join A to C and B to C to complete the triangle.

Justification: The construction is valid because the sum of any two sides of a triangle is greater than the third side (5 + 6 > 7, 5 + 7 > 6, 6 + 7 > 5). This ensures the triangle can be formed.

Question 17:
Construct a right-angled triangle PQR where ∠Q = 90°, PQ = 4 cm, and QR = 3 cm. Explain the steps and verify your construction.
Answer:

To construct a right-angled triangle PQR with the given measurements, follow these steps:


Step 1: Draw a horizontal line segment QR = 3 cm using a ruler.
Step 2: At point Q, construct a perpendicular line using a protractor or compass to ensure ∠Q = 90°.
Step 3: From Q, measure and mark PQ = 4 cm along the perpendicular line.
Step 4: Join P to R to complete the triangle.

Verification: Use the Pythagorean theorem to check if the triangle is right-angled. Here, PQ² + QR² = 4² + 3² = 16 + 9 = 25, and PR² (measured) should also be 25. Thus, PR = 5 cm, confirming the right angle at Q.

Question 18:
Construct a triangle ABC where AB = 6 cm, BC = 5 cm, and AC = 4 cm. Write the steps of construction clearly and justify each step.
Answer:

To construct triangle ABC with the given sides, follow these steps:


Step 1: Draw the base line
Draw a line segment AB of length 6 cm using a ruler.

Step 2: Mark point for side AC
With A as the center, draw an arc of radius 4 cm using a compass.

Step 3: Mark point for side BC
With B as the center, draw an arc of radius 5 cm using a compass. Ensure this arc intersects the previous arc drawn in Step 2.

Step 4: Complete the triangle
Label the point of intersection of the two arcs as C. Join A to C and B to C to form the triangle ABC.

Justification:
  • The arcs intersect only if the sum of any two sides is greater than the third side (Triangle Inequality Theorem).
  • Here, 4 + 5 > 6, 6 + 5 > 4, and 6 + 4 > 5, so the construction is valid.

Note: Ensure all measurements are accurate, and the arcs are drawn lightly for clarity. The final triangle should be neatly labeled with all sides and vertices marked.

Question 19:
Construct a triangle ABC where AB = 6 cm, BC = 5 cm, and ∠B = 60°. Also, draw the perpendicular bisector of side AC. Write the steps of construction clearly and justify your construction.
Answer:

To construct the triangle ABC and its perpendicular bisector of side AC, follow these steps:


Step 1: Draw the base BC = 5 cm
Use a ruler to draw a line segment BC of length 5 cm.

Step 2: Construct ∠B = 60° at point B
Place the protractor at point B and mark a point at 60°. Join this point to B to form a ray.

Step 3: Mark point A on the ray such that AB = 6 cm
From point B, measure 6 cm along the ray and mark point A.

Step 4: Join points A and C to complete the triangle ABC
Use a ruler to draw the line segment AC.

Step 5: Construct the perpendicular bisector of AC
  • With A as the center, draw arcs above and below AC using a compass (radius > half of AC).
  • Repeat the same with C as the center, ensuring the arcs intersect.
  • Join the intersection points to form the perpendicular bisector.

Justification: The perpendicular bisector passes through the midpoint of AC at 90°, ensuring equal distances from A and C. The triangle is accurately constructed using the given measurements.

Question 20:
Construct a triangle ABC where AB = 6 cm, BC = 5 cm, and ∠B = 60°. Also, draw a perpendicular bisector of side AC. Verify your construction by measuring the lengths and angles.
Answer:

To construct the triangle ABC and its perpendicular bisector, follow these steps:


Step 1: Draw the base AB
Draw a line segment AB of length 6 cm using a ruler.

Step 2: Construct ∠B = 60°
Place the protractor at point B and mark a point at 60°. Join this point to B to form a ray.

Step 3: Mark point C
From point B, measure 5 cm along the ray and mark point C.

Step 4: Complete the triangle
Join points A and C to complete the triangle ABC.

Step 5: Draw the perpendicular bisector of AC
Using a compass, draw arcs from points A and C that intersect above and below AC. Join these intersection points to form the perpendicular bisector.

Verification:
Measure AC and confirm the perpendicular bisector divides it into two equal parts. Also, verify that the angles and sides match the given conditions.

Note: Ensure all construction lines are light and neat, and the final triangle and bisector are clearly visible.

Question 21:
Construct a triangle ABC where AB = 6 cm, BC = 5 cm, and ∠B = 60°. Also, draw a perpendicular bisector of side AC. Write the steps of construction clearly and justify your construction.
Answer:

To construct the triangle ABC and its perpendicular bisector of side AC, follow these steps:


Step 1: Draw the base AB
Draw a line segment AB of length 6 cm using a ruler.

Step 2: Construct ∠B = 60°
At point B, use a protractor to draw an angle of 60°. Mark a point X on the angle's arm.

Step 3: Mark point C
From B, measure 5 cm along the angle's arm and mark point C.

Step 4: Complete the triangle
Join points A and C to complete the triangle ABC.

Step 5: Draw the perpendicular bisector of AC
Using a compass, draw arcs from A and C that intersect above and below AC. Join these intersection points to form the perpendicular bisector.

Justification: The perpendicular bisector passes through the midpoint of AC at a right angle, ensuring equal distances from A and C.

Question 22:
Construct a quadrilateral PQRS where PQ = 4.5 cm, QR = 5 cm, RS = 4 cm, SP = 5.5 cm, and diagonal PR = 6 cm. Explain each step of the construction and verify the properties of the quadrilateral.
Answer:

To construct quadrilateral PQRS, follow these steps:


Step 1: Draw diagonal PR
Draw a line segment PR of length 6 cm.

Step 2: Construct triangle PQR
With P as the center, draw an arc of radius 4.5 cm (for PQ).
With R as the center, draw an arc of radius 5 cm (for QR).
The intersection of these arcs is point Q. Join P to Q and Q to R.

Step 3: Construct triangle PRS
With P as the center, draw an arc of radius 5.5 cm (for SP).
With R as the center, draw an arc of radius 4 cm (for RS).
The intersection of these arcs is point S. Join P to S and S to R.

Verification:
  • Measure all sides to confirm they match the given lengths.
  • Check that the sum of angles in the quadrilateral is 360°.
Question 23:
Construct a triangle ABC where AB = 6 cm, ∠B = 45°, and BC = 5 cm. Write the steps of construction and justify each step.
Answer:

To construct the triangle ABC with the given measurements, follow these steps:


Step 1: Draw the base line segment BC = 5 cm using a ruler.
Justification: The base of the triangle is fixed as per the given measurement.

Step 2: At point B, construct an angle of 45° using a protractor.
Justification: This ensures that ∠B is accurately measured.

Step 3: From point B, measure 6 cm along the new line to mark point A.
Justification: This ensures the side AB is exactly 6 cm.

Step 4: Join point A to point C to complete the triangle.
Justification: The third side is drawn to close the triangle.

Verification: Measure all sides and angles to ensure they match the given values. The triangle should satisfy the Angle-Side-Angle (ASA) criterion for uniqueness.

Question 24:
Construct a quadrilateral PQRS where PQ = 4 cm, QR = 5 cm, RS = 4.5 cm, SP = 5.5 cm, and ∠P = 60°. Explain the steps and verify your construction.
Answer:

To construct quadrilateral PQRS with the given measurements, follow these steps:


Step 1: Draw side PQ = 4 cm using a ruler.
Justification: The first side is fixed as per the given length.

Step 2: At point P, construct an angle of 60° using a protractor.
Justification: This ensures ∠P is accurately measured.

Step 3: From point P, measure 5.5 cm along the new line to mark point S.
Justification: This ensures side SP is exactly 5.5 cm.

Step 4: From point Q, draw an arc of radius 5 cm to locate point R.
Justification: This ensures side QR is 5 cm.

Step 5: From point S, draw an arc of radius 4.5 cm to intersect the previous arc at point R.
Justification: This ensures side RS is 4.5 cm.

Step 6: Join points Q to R and R to S to complete the quadrilateral.
Justification: All sides and angles are now correctly connected.

Verification: Measure all sides and angles to confirm they match the given values. The quadrilateral should satisfy the Side-Angle-Side-Side (SASS) criterion for uniqueness.

Case-based Questions (4 Marks) – with Solutions (CBSE Pattern)

These 4-mark case-based questions assess analytical skills through real-life scenarios. Answers must be based on the case study provided.

Question 1:
A student needs to construct a triangle ABC where AB = 5 cm, BC = 6 cm, and ∠B = 60°. Explain the steps involved in this construction.
Answer:
Problem Interpretation

We need to construct a triangle with two sides and the included angle given, as per our textbook.

Mathematical Modeling
  • Draw base BC = 6 cm.
  • At point B, construct ∠B = 60° using a protractor.
Solution
  • Mark point A on the new line such that AB = 5 cm.
  • Join A to C to complete ΔABC. [Diagram: Triangle ABC with labeled sides and angle]
Question 2:
Construct a right-angled triangle PQR where PQ = 4 cm, PR = 3 cm, and ∠P = 90°. Justify your steps.
Answer:
Problem Interpretation

We studied constructing right-angled triangles when two sides including the right angle are given.

Mathematical Modeling
  • Draw PQ = 4 cm.
  • At P, construct ∠P = 90° using a set square.
Solution
  • Mark R such that PR = 3 cm on the perpendicular line.
  • Join R to Q to form ΔPQR. [Diagram: Right-angled triangle PQR with labeled sides]
Question 3:
A student needs to construct a triangle ABC where AB = 5 cm, BC = 6 cm, and ∠B = 60°. Explain the steps involved and justify the construction.
Answer:
Problem Interpretation

We need to construct a triangle with two sides and the included angle given, as per NCERT examples.

Mathematical Modeling
  • Draw base BC = 6 cm.
  • At B, construct ∠B = 60° using a protractor.
  • Mark A on the new arm such that BA = 5 cm.
  • Join AC to complete ΔABC.
Solution

This follows the SAS criterion, ensuring uniqueness. [Diagram: Triangle ABC with labeled sides and angle.]

Question 4:
A gardener wants to divide a circular flower bed into 6 equal parts using compass and ruler. Describe the construction process.
Answer:
Problem Interpretation

We studied dividing a circle into equal parts using central angles, a real-life application.

Mathematical Modeling
  • Draw a circle of any radius.
  • Mark center O and draw a radius.
  • Using a protractor, mark 60° angles (360°/6) around O.
  • Join points to divide the circle into 6 equal sectors.
Solution

Each sector has a central angle of 60°, ensuring equal division. [Diagram: Circle divided into 6 equal parts.]

Question 5:
A student needs to construct a triangle ABC where AB = 5 cm, BC = 6 cm, and ∠B = 60°. Explain the steps using ruler and compass and justify the construction.
Answer:
Problem Interpretation

We need to construct a triangle with two sides and an included angle using ruler and compass.

Mathematical Modeling

Our textbook shows that such constructions require drawing the angle first, then marking the sides.

Solution
  • Draw base BC = 6 cm.
  • At B, construct ∠B = 60° using protractor.
  • Mark A on the new arm 5 cm from B.
  • Join AC to complete ΔABC.
[Diagram: Triangle ABC with labeled sides and angle]
Question 6:
Construct a right-angled triangle PQR where PQ = 4 cm, PR = 3 cm, and ∠P = 90°. List the steps and verify using the Pythagoras theorem.
Answer:
Problem Interpretation

We must construct a right-angled triangle given two sides forming the right angle.

Mathematical Modeling

Our textbook demonstrates constructing right angles using perpendicular lines.

Solution
  • Draw PQ = 4 cm.
  • At P, construct a perpendicular using compass.
  • Mark R on the perpendicular 3 cm from P.
  • Join QR to complete ΔPQR.

Verification: By Pythagoras, 3² + 4² = 5². Measuring QR confirms ≈5 cm.

[Diagram: Right-angled triangle PQR]
Question 7:
A student needs to construct a triangle ABC where AB = 5 cm, ∠A = 60°, and AC = 7 cm. Explain the steps using ruler and compass and justify the construction.
Answer:
Problem Interpretation

We need to construct a triangle with given sides and angle using ruler and compass.

Mathematical Modeling
  • Draw base AB = 5 cm.
  • At point A, construct ∠A = 60° using protractor.
  • Mark AC = 7 cm on the angle line.
  • Join B and C to complete ΔABC.
Solution

Our textbook shows that this method ensures accuracy. The triangle is valid as the sum of two sides (5 + 7) > third side.

Question 8:
Construct a right-angled triangle PQR where ∠Q = 90°, PQ = 4 cm, and QR = 3 cm. Verify using the Pythagoras theorem.
Answer:
Problem Interpretation

We must construct a right-angled triangle with given legs and verify using Pythagoras theorem.

Mathematical Modeling
  • Draw PQ = 4 cm.
  • At Q, construct ∠Q = 90°.
  • Mark QR = 3 cm perpendicular to PQ.
  • Join PR to complete ΔPQR.
Solution

We studied that PR² = PQ² + QR² = 16 + 9 = 25 ⇒ PR = 5 cm. Measuring PR confirms the construction.

Question 9:
Construct a right-angled triangle PQR where PQ = 4 cm, PR = 3 cm, and ∠P = 90°. List the steps and verify the construction.
Answer:
Problem Interpretation

We must construct a right-angled triangle with given legs using geometric tools.

Mathematical Modeling
  • Draw PQ = 4 cm.
  • At P, construct ∠P = 90° using a protractor.
  • Mark R on the perpendicular arm such that PR = 3 cm.
  • Join QR to form ΔPQR.
Solution

We studied that this satisfies Pythagoras' theorem (3² + 4² = 5²). The hypotenuse QR should measure 5 cm for validation.

Question 10:
A student needs to construct a triangle ABC where AB = 5 cm, BC = 6 cm, and ∠B = 60°. Explain the steps using ruler and compass.
Answer:
Problem Interpretation

We need to construct a triangle with given sides and included angle.

Mathematical Modeling
  • Draw base BC = 6 cm.
  • At B, construct ∠B = 60° using a protractor.
  • Mark A on the new arm such that AB = 5 cm.
  • Join AC to complete ΔABC.
Solution

Our textbook shows similar constructions. Verify lengths with a ruler and angle with a protractor.

Question 11:
Construct a right-angled triangle PQR where ∠Q = 90°, PQ = 4 cm, and PR = 5 cm. Verify using the Pythagoras theorem.
Answer:
Problem Interpretation

We must construct a right-angled triangle with one leg and hypotenuse given, a common NCERT problem.


Mathematical Modeling
  • Draw PQ = 4 cm.
  • At Q, construct ∠Q = 90°.
  • From P, draw an arc of 5 cm to meet the perpendicular at R.

Solution

Verification: QR = 3 cm (since 3² + 4² = 5²). Our textbook uses Pythagoras theorem to validate such constructions.

Question 12:
Construct a right-angled triangle PQR where PQ = 4 cm, PR = 3 cm, and ∠P = 90°. Verify using the Pythagoras theorem.
Answer:
Problem Interpretation

Construct a right-angled triangle with given legs and verify using Pythagoras theorem.

Mathematical Modeling
  • Draw PQ = 4 cm.
  • At P, construct ∠P = 90°.
  • Mark R on the perpendicular arm 3 cm from P.
  • Join QR.
Solution

We studied that 3² + 4² = 5², so QR must be 5 cm. Measuring QR confirms the construction.

[Diagram: Right-angled triangle PQR with labeled sides]
Question 13:
A student needs to construct a triangle ABC where AB = 5 cm, ∠A = 60°, and AC = 7 cm. Explain the steps using ruler and compass.
Answer:
Problem Interpretation

We need to construct a triangle with given sides and angle using ruler and compass.

Mathematical Modeling
  • Draw base AB = 5 cm.
  • At A, construct ∠A = 60° using protractor.
  • Mark AC = 7 cm on the angle line.
  • Join B and C to complete ΔABC.
Solution

Our textbook shows similar constructions. Ensure measurements are precise for accuracy.

Question 14:
Construct a right-angled triangle PQR where ∠Q = 90°, PQ = 4 cm, and QR = 3 cm. Verify using Pythagoras theorem.
Answer:
Problem Interpretation

Construct a right-angled triangle with given legs and verify using Pythagoras theorem.

Mathematical Modeling
  • Draw PQ = 4 cm.
  • At Q, draw a perpendicular using compass.
  • Mark QR = 3 cm on the perpendicular.
  • Join PR to complete ΔPQR.
Solution

We studied that PR² = PQ² + QR² = 16 + 9 = 25. Thus, PR = 5 cm.

Question 15:
A student needs to construct a triangle ABC where AB = 5 cm, BC = 6 cm, and ∠B = 60°. Explain the steps involved using a ruler and compass.
Answer:
Problem Interpretation

We need to construct a triangle with two sides and the included angle given.


Mathematical Modeling
  • Draw base BC = 6 cm.
  • At point B, construct ∠B = 60° using a protractor.

Solution

From B, mark AB = 5 cm on the angle arm. Join A to C to complete ΔABC. [Diagram: Triangle with labeled sides and angle]

Question 16:
Construct a right-angled triangle PQR where ∠Q = 90°, PQ = 4 cm, and PR = 5 cm. List the steps and verify using the Pythagoras theorem.
Answer:
Problem Interpretation

We must construct a right-angled triangle with one leg and hypotenuse given.

Mathematical Modeling

We studied that Pythagoras theorem helps verify such constructions.

Solution
  • Draw PQ = 4 cm and construct ∠Q = 90°.
  • From P, draw an arc of radius 5 cm to meet the perpendicular at R.
  • Join R to Q and measure QR = 3 cm.
  • Verify: 4² + 3² = 5².
Question 17:
A student needs to construct a triangle ABC where AB = 5 cm, ∠A = 60°, and AC = 4 cm. Explain the steps involved and justify the construction.
Answer:
Problem Interpretation

We need to construct a triangle with given sides and angle using ruler and compass.

Mathematical Modeling
  • Draw base AB = 5 cm
  • At A, construct ∠A = 60° using protractor
  • Mark C on the new arm 4 cm from A
  • Join BC to complete ΔABC
Solution

Our textbook shows this method ensures unique triangle formation by SAS congruence rule. The construction is valid as sides and angle satisfy triangle inequality.

Question 18:
Construct a right-angled triangle PQR where ∠Q = 90°, PQ = 3 cm, and PR = 5 cm. Verify your construction using Pythagoras theorem.
Answer:
Problem Interpretation

We must construct a right triangle with given legs and hypotenuse.

Mathematical Modeling
  • Draw PQ = 3 cm at 90° to QR
  • From P, draw arc of radius 5 cm
  • Intersection gives point R
  • Join QR to complete ΔPQR
Solution

We studied that 3² + QR² = 5² ⇒ QR = 4 cm. Measurement confirms QR ≈ 4 cm, validating our construction through Pythagoras theorem.

Question 19:
Construct a perpendicular bisector of a line segment PQ of length 8 cm. Justify your steps.
Answer:
Problem Interpretation

We studied how to draw a perpendicular bisector for a given line segment.

Mathematical Modeling

Our textbook uses the concept of equidistant points from P and Q.

Solution
  • Draw PQ = 8 cm.
  • With P and Q as centers, draw arcs of radius > 4 cm intersecting above and below PQ.
  • Join the intersection points to get the perpendicular bisector.
[Diagram: Line PQ with perpendicular bisector]
Question 20:
A student is asked to construct a triangle ABC where AB = 5 cm, ∠A = 60°, and ∠B = 45°. Explain the step-by-step construction process and justify each step.
Answer:

To construct △ABC with the given measurements, follow these steps:


Step 1: Draw the base line segment AB = 5 cm using a ruler.
Step 2: At point A, construct ∠A = 60° using a protractor or compass.
Step 3: At point B, construct ∠B = 45° using the same method.
Step 4: The two constructed rays from A and B will intersect at point C, completing the triangle.

Justification: The sum of angles in a triangle is 180°, so ∠C = 180° - (60° + 45°) = 75°. The construction ensures all sides and angles are accurately represented.

Question 21:
A farmer wants to divide his triangular field PQR into two equal parts by constructing a perpendicular from vertex P to the opposite side QR. Given PQ = 6 cm, PR = 8 cm, and QR = 10 cm, describe the construction steps and verify the division of area.
Answer:

To divide △PQR into two equal areas, follow these steps:


Step 1: Draw △PQR with sides PQ = 6 cm, PR = 8 cm, and QR = 10 cm using a ruler and compass.
Step 2: Construct the perpendicular bisector from vertex P to side QR using compass arcs.
Step 3: Mark the foot of the perpendicular as point S, dividing QR into two segments.

Verification: The area of △PQR is (1/2) × base × height. Since PS divides the triangle into two smaller triangles △PQS and △PRS with equal heights and equal base segments (as PS is perpendicular), their areas are equal.

Question 22:
A student is asked to construct a triangle ABC where AB = 5 cm, BC = 6 cm, and ∠B = 60°. Explain the step-by-step construction process and justify each step.
Answer:

To construct triangle ABC with the given measurements, follow these steps:


Step 1: Draw the base line segment BC = 6 cm using a ruler.
Step 2: At point B, use a protractor to construct an angle of 60° and mark it.
Step 3: From point B, measure 5 cm along the angle line to locate point A.
Step 4: Join points A and C to complete the triangle.

Justification: The construction ensures all given conditions are met—AB = 5 cm, BC = 6 cm, and ∠B = 60°. The triangle is uniquely determined by these measurements.

Question 23:
A teacher asks students to construct a perpendicular bisector of a line segment PQ of length 8 cm. Describe the construction process and explain why this method works.
Answer:

To construct the perpendicular bisector of PQ = 8 cm, follow these steps:


Step 1: Draw line segment PQ = 8 cm using a ruler.
Step 2: With P as the center, draw arcs above and below the line using a compass (radius > 4 cm).
Step 3: Repeat step 2 with Q as the center, ensuring the arcs intersect.
Step 4: Join the intersection points of the arcs to form the perpendicular bisector.

Explanation: The method works because the perpendicular bisector is the locus of points equidistant from both P and Q. The intersecting arcs ensure equal distances, guaranteeing a perfect right angle at the midpoint.

Question 24:
Rahul wants to construct a triangle ABC where AB = 5 cm, BC = 6 cm, and ∠B = 60°. However, he mistakenly draws ∠B = 50° instead of 60°.

Explain the steps Rahul should follow to correct his construction and also justify why the initial attempt was incorrect.

Answer:

To correct the construction, Rahul should follow these steps:


Step 1: Draw the base line segment BC = 6 cm using a ruler.
Step 2: At point B, use a protractor to measure and mark the correct angle of 60° (not 50°).
Step 3: From point B, measure 5 cm along the new angle line to locate point A.
Step 4: Join points A and C to complete the triangle ABC.

Justification: The initial attempt was incorrect because the angle at B was 50° instead of 60°, which would result in a triangle with different side lengths and angles than required. The properties of a triangle depend on both sides and the included angle, so an incorrect angle leads to an incorrect shape.

Question 25:
Priya is constructing a perpendicular bisector of a line segment XY = 8 cm. She uses a compass to draw arcs from points X and Y, but the arcs do not intersect.

Identify the possible mistake Priya made and describe the correct method to construct the perpendicular bisector.

Answer:

Possible Mistake: Priya might have used a compass opening less than half the length of XY, causing the arcs to not intersect. The compass width must be greater than 4 cm (half of 8 cm) for the arcs to intersect.


Correct Method:
Step 1: Draw line segment XY = 8 cm.
Step 2: Set the compass width to more than 4 cm (e.g., 5 cm).
Step 3: Draw arcs from both X and Y above and below the line.
Step 4: The arcs will intersect at two points. Join these points to form the perpendicular bisector.

Note: The perpendicular bisector divides XY into two equal parts of 4 cm each and forms a 90° angle with XY.

Question 26:
A student is asked to construct a triangle ABC where AB = 5 cm, BC = 6 cm, and ∠B = 60°. However, the student mistakenly draws ∠B as 45° instead of 60°.

Explain the steps to correct this construction and justify why the initial attempt was incorrect.

Answer:

To correct the construction, follow these steps:


Step 1: Draw the base line segment BC = 6 cm using a ruler.
Step 2: At point B, use a protractor to measure and mark the correct angle of 60° (not 45°).
Step 3: From point B, measure 5 cm along the new angle line to locate point A.
Step 4: Join points A and C to complete the triangle ABC.

The initial attempt was incorrect because the angle at B was 45° instead of 60°, which would result in a triangle with different side lengths and angles, violating the given conditions. The correct angle ensures the triangle's sides and angles match the required measurements.

Question 27:
A teacher asks students to construct a perpendicular bisector of a line segment PQ = 8 cm. One student uses a compass but forgets to adjust its width properly.

Describe the correct procedure to construct the perpendicular bisector and explain the importance of adjusting the compass width.

Answer:

The correct procedure to construct the perpendicular bisector of PQ is:


Step 1: Draw line segment PQ = 8 cm using a ruler.
Step 2: Adjust the compass width to slightly more than half of PQ (i.e., > 4 cm).
Step 3: With P as the center, draw arcs above and below PQ.
Step 4: Repeat Step 3 with Q as the center, ensuring the arcs intersect.
Step 5: Join the intersection points of the arcs to form the perpendicular bisector.

Adjusting the compass width is crucial because if it is too small, the arcs won't intersect, and if it is too large, the construction becomes inaccurate. The correct width ensures precise intersection points for the perpendicular bisector.

Question 28:
A student is asked to construct a triangle ABC where AB = 5 cm, BC = 6 cm, and ∠B = 60°. However, the student mistakenly draws ∠B = 50° instead of 60°. Explain the steps to correct this construction and justify why the initial attempt was incorrect.
Answer:

To correct the construction, follow these steps:


Step 1: Draw the base BC = 6 cm using a ruler.
Step 2: At point B, use a protractor to measure the correct angle of 60° (not 50°).
Step 3: Mark a point A on the new arm of the angle such that AB = 5 cm.
Step 4: Join A to C to complete the triangle ABC.

The initial attempt was incorrect because the angle at B was 50°, which does not match the given condition of 60°. This error would result in a triangle with incorrect side lengths and angles, violating the SSA (Side-Side-Angle) condition for unique triangle construction.

Question 29:
A teacher asks students to construct a quadrilateral PQRS where PQ = 4 cm, QR = 5 cm, RS = 4.5 cm, SP = 5.5 cm, and diagonal PR = 6 cm. One student constructs triangles PQR and PRS separately but struggles to join them. Explain the correct procedure and the geometric principle ensuring the quadrilateral's uniqueness.
Answer:

The correct construction steps are:


Step 1: Draw triangle PQR using PQ = 4 cm, QR = 5 cm, and PR = 6 cm (by SSS construction).
Step 2: From point R, measure RS = 4.5 cm using a compass.
Step 3: From point P, measure PS = 5.5 cm using a compass.
Step 4: The intersection of these arcs will give point S, completing the quadrilateral.

The geometric principle ensuring uniqueness is the SSS congruency of triangles PQR and PRS. Since all sides are fixed, the quadrilateral is uniquely determined. The student's error likely arose from not ensuring the exact measurements or overlapping arcs for point S.

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