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Constructions – CBSE NCERT Study Resources

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Study Materials

10th

10th - 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 division of line segments, construction of triangles under given conditions, and tangents to circles. These concepts are essential for developing spatial reasoning and problem-solving skills in geometry.

Geometric Construction: The process of drawing precise geometric figures using only a compass and an unmarked straightedge, without measurements.

Key Topics Covered

  • Division of a Line Segment in a Given Ratio
  • Construction of Similar Triangles
  • Construction of Tangents to a Circle

Division of a Line Segment in a Given Ratio

This section explains how to divide a line segment internally in a specified ratio using the basic proportionality theorem and construction techniques.

Basic Proportionality Theorem (Thales' Theorem): If a line is drawn parallel to one side of a triangle intersecting the other two sides, it divides those sides proportionally.

Construction of Similar Triangles

Students learn to construct triangles similar to a given triangle with a specified scale factor. This involves understanding the properties of similar figures and applying them in constructions.

Construction of Tangents to a Circle

This topic covers the construction of tangents to a circle from an external point, emphasizing the perpendicularity between the radius and the tangent at the point of contact.

Tangent to a Circle: A line that touches the circle at exactly one point, perpendicular to the radius at that point.

Summary

The chapter equips students with practical skills in geometric constructions, reinforcing theoretical concepts through hands-on activities. Mastery of these techniques is crucial for solving complex geometric problems in higher grades.

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 first step in constructing a triangle with sides 5 cm, 6 cm, and 7 cm?
Answer:
Draw base line of 7 cm.
Question 2:
How many arcs are drawn to construct a 60° angle?
Answer:
2 arcs (one for radius, one for angle).
Question 3:
What instrument is used to draw perpendicular bisector of a line segment?
Answer:
Compass and ruler.
Question 4:
In constructing a tangent to a circle, what angle is formed at the point of contact?
Answer:
90° (right angle).
Question 5:
What is the length of the altitude in an equilateral triangle of side 4 cm?
Answer:
2√3 cm.
Question 6:
How many parallel lines can be drawn through a point not on the given line?
Answer:
Only one.
Question 7:
What is the scale factor if a 3 cm line is reduced to 1 cm?
Answer:
1/3.
Question 8:
Which theorem is used to divide a line segment in a given ratio?
Answer:
Basic Proportionality Theorem.
Question 9:
What is constructed first in a right-angled triangle with hypotenuse 5 cm?
Answer:
Hypotenuse as base line.
Question 10:
How many circles pass through three non-collinear points?
Answer:
Only one.
Question 11:
What is the radius used to draw an angle bisector?
Answer:
Any convenient radius.
Question 12:
What is the minimum number of measurements needed to construct a unique quadrilateral?
Answer:
5 measurements.
Question 13:
What is the first step in constructing a triangle when its base, a base angle, and the sum of the other two sides are given?
Answer:

The first step is to draw the base of the triangle using the given length.
Then, at one endpoint of the base, construct the given base angle using a protractor.

Question 14:
How do you bisect a given line segment using a compass and ruler?
Answer:

To bisect a line segment:
1. Draw arcs from both ends of the segment with a radius greater than half its length.
2. The intersecting points of these arcs will help draw the perpendicular bisector.

Question 15:
What is the purpose of constructing a tangent to a circle from an external point?
Answer:

The purpose is to draw a line that touches the circle at exactly one point from an outside point.
This is useful in geometry problems involving circles and their properties.

Question 16:
Explain how to construct a triangle with a given perimeter and two base angles.
Answer:

1. Draw a line segment equal to the given perimeter.
2. Construct the two given base angles at the endpoints.
3. The intersection of the angle lines forms the third vertex of the triangle.

Question 17:
What is the significance of the angle bisector in constructions?
Answer:

The angle bisector divides an angle into two equal parts.
It is crucial for ensuring symmetry and accuracy in geometric constructions.

Question 18:
How do you construct a right-angled triangle given its hypotenuse and one side?
Answer:

1. Draw the given side as one leg.
2. At one end, construct a perpendicular.
3. From the other end, draw an arc with the hypotenuse length to intersect the perpendicular, forming the triangle.

Question 19:
What is the role of the circumradius in constructing a triangle?
Answer:

The circumradius is the radius of the circumscribed circle around the triangle.
It helps in constructing triangles when the vertices must lie on a specific circle.

Question 20:
Describe the steps to construct a quadrilateral when four sides and one diagonal are given.
Answer:

1. Draw the given diagonal.
2. From each endpoint, draw arcs with the lengths of the adjacent sides.
3. The intersection points of these arcs will form the other two vertices.

Question 21:
Why is it important to keep the compass width unchanged during certain constructions?
Answer:

Keeping the compass width unchanged ensures that all arcs drawn have the same radius.
This is crucial for maintaining accuracy in geometric constructions like bisectors or perpendiculars.

Question 22:
How do you construct a tangent to a circle at a point on it without using the center?
Answer:

1. Draw any chord through the given point.
2. Construct the perpendicular to the chord at that point.
3. This perpendicular line is the required tangent.

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 a triangle ABC where AB = 5 cm, BC = 6 cm, and ∠B = 60°. Write the steps involved.
Answer:

To construct triangle ABC with given measurements:

  • Draw base BC = 6 cm.
  • At point B, construct ∠B = 60° using a protractor.
  • From B, mark AB = 5 cm on the new line.
  • Join A to C to complete the triangle.

Ensure all sides and angles are accurately measured for correctness.

Question 2:
How do you construct a perpendicular bisector of a line segment of length 8 cm?
Answer:

Steps to construct the perpendicular bisector of an 8 cm line segment:

  • Draw AB = 8 cm.
  • With A as center, draw arcs above and below AB using a compass (radius > 4 cm).
  • Repeat with B as center, intersecting the previous arcs.
  • Join the intersection points to form the perpendicular bisector.

The bisector divides AB into two equal parts at 90°.

Question 3:
Construct a triangle PQR with PQ = 7 cm, QR = 5 cm, and PR = 6 cm. List the steps.
Answer:

Construction steps for triangle PQR:

  • Draw base PQ = 7 cm.
  • From P, draw an arc with radius = 6 cm (PR).
  • From Q, draw an arc with radius = 5 cm (QR).
  • Join the intersection point R to P and Q.

Verify all sides match the given measurements.

Question 4:
Explain how to construct an angle of 45° using a compass and ruler.
Answer:

To construct a 45° angle:

  • First, construct a 90° angle by bisecting a straight line.
  • Bisect the 90° angle to get two 45° angles.
  • Use the compass to mark equal arcs and draw the bisector.

This ensures precision in creating the angle.

Question 5:
Construct a triangle XYZ where XY = 4 cm, YZ = 5 cm, and ∠Y = 90°. Describe the steps.
Answer:

Steps to construct right-angled triangle XYZ:

  • Draw XY = 4 cm.
  • At Y, construct ∠Y = 90° using a protractor.
  • From Y, mark YZ = 5 cm on the perpendicular line.
  • Join X to Z to complete the triangle.

Check the right angle and side lengths for accuracy.

Question 6:
How would you construct a line segment of 3.5 cm using a ruler and compass?
Answer:

To construct a 3.5 cm line segment:

  • Place the compass at the zero mark of the ruler.
  • Adjust the compass to 3.5 cm.
  • Draw a line segment by marking the endpoint with the compass.

Measure the segment to confirm its length.

Question 7:
Construct a triangle DEF with DE = 6 cm, ∠D = 60°, and ∠E = 45°. Outline the steps.
Answer:

Steps to construct triangle DEF:

  • Draw DE = 6 cm.
  • At D, construct ∠D = 60°.
  • At E, construct ∠E = 45°.
  • Extend the arms to meet at F.

Ensure the angles are accurately measured for correctness.

Question 8:
Describe the construction of an equilateral triangle with each side measuring 5 cm.
Answer:

To construct an equilateral triangle:

  • Draw base AB = 5 cm.
  • From A, draw an arc with radius = 5 cm.
  • From B, draw another arc intersecting the first.
  • Join the intersection point to A and B.

All sides and angles (60°) will be equal.

Question 9:
Construct a triangle LMN where LM = 4 cm, MN = 5 cm, and ∠M = 30°. List the steps.
Answer:

Steps to construct triangle LMN:

  • Draw LM = 4 cm.
  • At M, construct ∠M = 30°.
  • From M, mark MN = 5 cm on the new line.
  • Join L to N to complete the triangle.

Verify the angle and side lengths for accuracy.

Question 10:
How do you bisect a given angle of 80° using a compass and ruler?
Answer:

To bisect an 80° angle:

  • Draw the given angle using a protractor.
  • From the vertex, draw arcs intersecting the angle's arms.
  • From these points, draw intersecting arcs inside the angle.
  • Join the vertex to the intersection point to form the bisector.

The bisector divides the angle into two 40° angles.

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 ABC with given sides:


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

Verification: Measure the sides to ensure they match the given lengths.

Question 2:
Construct a tangent to a circle of radius 4 cm from a point 8 cm away from its center. Explain the steps.
Answer:

Steps to construct the tangent:


1. Draw a circle with center O and radius 4 cm.
2. Mark a point P 8 cm away from O.
3. Join OP and draw its perpendicular bisector to find its midpoint M.
4. With M as center and radius OM, draw a semicircle intersecting the circle at T.
5. Join PT, which is the required tangent.

Reasoning: The angle between the tangent and radius is 90°, so OTP is a right angle.

Question 3:
Construct a pair of tangents to a circle of radius 3 cm inclined at 60° to each other. Describe the construction process.
Answer:

Construction steps:


1. Draw a circle with center O and radius 3 cm.
2. Draw two radii OA and OB such that ∠AOB = 120° (since tangents are inclined at 60°, the angle between radii is 180° - 60° = 120°).
3. At points A and B, draw perpendiculars to OA and OB, respectively.
4. The perpendiculars intersect at point P outside the circle, forming the tangents PA and PB.

Note: The angle between the tangents is 60° as required.

Question 4:
Construct a triangle PQR with PQ = 6 cm, ∠Q = 45°, and ∠R = 60°. List the steps.
Answer:

Steps to construct PQR:


1. Draw base PQ = 6 cm.
2. At point Q, construct a 45° angle using a protractor.
3. The sum of angles in a triangle is 180°, so ∠P = 180° - (45° + 60°) = 75°.
4. At point P, construct a 75° angle.
5. The point where the two angle lines meet is R, completing the triangle.

Check: Measure the angles to ensure they match the given values.

Question 5:
Divide a line segment AB of length 8 cm into 4 equal parts using compass and ruler. Explain the method.
Answer:

Steps to divide AB into 4 equal parts:


1. Draw AB = 8 cm.
2. From A, draw an acute angle line AX using a ruler.
3. Using a compass, mark 4 equal arcs of any convenient length on AX, naming the points as C, D, E, and F.
4. Join FB.
5. Draw lines parallel to FB through C, D, and E intersecting AB at points P, Q, and R.

Result: AP = PQ = QR = RB = 2 cm each.

Question 6:
Construct a triangle PQR where PQ = 6 cm, ∠Q = 45°, and PR + QR = 9 cm. Write the steps clearly.
Answer:

To construct PQR with given conditions:


Step 1: Draw base PQ = 6 cm.
Step 2: At Q, construct ∠PQR = 45°.
Step 3: Extend QP to S such that QS = 9 cm (since PR + QR = 9 cm).
Step 4: Join S to R (where R lies on the angle arm).
Step 5: Draw the perpendicular bisector of SR to intersect QS at R.

Key Point: The sum condition ensures R lies on the locus satisfying PR + QR = 9 cm.

Question 7:
Construct a quadrilateral ABCD where AB = 4 cm, BC = 5 cm, CD = 6 cm, ∠B = 105°, and ∠C = 80°. Explain the steps.
Answer:

To construct quadrilateral ABCD:


Step 1: Draw AB = 4 cm.
Step 2: At B, construct ∠ABC = 105° and mark BC = 5 cm.
Step 3: At C, construct ∠BCD = 80° and mark CD = 6 cm.
Step 4: Join A to D to complete the quadrilateral.

Verification: Measure all sides and angles to ensure accuracy. Use a protractor for angle precision.

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 BC = 7 cm, ∠B = 45°, and AB + AC = 13 cm. Justify your steps.
Answer:
Introduction
We studied constructing triangles given base, angle, and sum of two sides.
Argument 1
1. Draw BC = 7 cm. At B, draw ∠B = 45°.
2. Mark D on BX such that BD = 13 cm (AB + AC). Join D to C.
Argument 2
3. Draw perpendicular bisector of DC meeting BD at A.
4. Join A to C. ABC is the required triangle.
Conclusion
Our textbook shows this method ensures AB + AC = BD = 13 cm.
Question 2:
Construct a tangent to a circle of radius 4 cm from a point 8 cm away from its center. Explain the steps.
Answer:
Introduction
We learned to draw tangents from an external point to a circle.
Argument 1
1. Draw circle with O (center) and radius 4 cm. Mark P, 8 cm away.
2. Join OP and draw its perpendicular bisector to find midpoint M.
Argument 2
3. Draw circle with M as center and MO as radius. It intersects the first circle at T.
4. PT is the tangent.
Conclusion
Our NCERT example proves PT is perpendicular to OT, satisfying tangent properties.
Question 3:
Construct a triangle PQR with QR = 6 cm, ∠Q = 60°, and PQ - PR = 2 cm. Validate your construction.
Answer:
Introduction
We practiced constructing triangles when side difference is given.
Argument 1
1. Draw QR = 6 cm. At Q, draw ∠XQR = 60°.
2. Mark S on QX such that QS = 2 cm (PQ - PR). Join S to R.
Argument 2
3. Draw perpendicular bisector of SR meeting QX at P.
4. Join P to R. PQR is the required triangle.
Conclusion
As per NCERT, PQ - PR = QS = 2 cm is verified by measurement.
Question 4:
Construct a pair of tangents to a circle of radius 5 cm inclined at 60° to each other. Describe the steps.
Answer:
Introduction
We studied drawing tangents inclined at a given angle.
Argument 1
1. Draw circle with O (radius = 5 cm). Draw radii OA and OB making ∠AOB = 120° (180° - 60°).
Argument 2
2. At A and B, draw perpendiculars to OA and OB respectively, intersecting at P.
3. PA and PB are the required tangents.
Conclusion
Our textbook confirms ∠APB = 60° as tangents are equally inclined.
Question 5:
Construct a triangle XYZ with YZ = 5 cm, ∠Y = 30°, and XY + XZ = 8 cm. Justify the construction.
Answer:
Introduction
We learned to construct triangles with base, angle, and side sum.
Argument 1
1. Draw YZ = 5 cm. At Y, draw ∠PYZ = 30°.
2. Mark W on YP such that YW = 8 cm (XY + XZ). Join W to Z.
Argument 2
3. Draw perpendicular bisector of WZ meeting YP at X.
4. Join X to Z. XYZ is the required triangle.
Conclusion
As per NCERT, XY + XZ = YW = 8 cm is verified by measurement.
Question 6:
Construct a triangle ABC where BC = 7 cm, ∠B = 45°, and AB + AC = 13 cm. Justify your steps and verify the construction.
Answer:
Introduction

We studied constructing triangles given base, angle, and sum of two sides. Our textbook shows similar problems.


Argument 1
  • Draw BC = 7 cm and mark ∠B = 45°.
  • Extend BX and cut BD = 13 cm (AB + AC).
  • Join D to C and draw perpendicular bisector of DC.

Argument 2
  • Where bisector meets BD, mark A.
  • Join A to C to complete ΔABC.
  • Measure AB + AC to verify = 13 cm.

Conclusion

This method ensures accuracy. Real-life uses include designing structures.

Question 7:
Construct a tangent to a circle of radius 4 cm from a point 10 cm away from its center. Explain steps with justification.
Answer:
Introduction

We learned to draw tangents from external points. NCERT has similar examples.


Argument 1
  • Draw circle (O, 4 cm) and mark P 10 cm away.
  • Join OP and bisect it to find midpoint M.

Argument 2
  • Draw circle (M, MO) intersecting given circle at T.
  • Join PT, which is the required tangent.
  • Verify ∠OTP = 90° using protractor.

Conclusion

This follows the theorem. Applications include road curves.

Question 8:
Construct a triangle ABC where AB = 5 cm, ∠A = 60°, and ∠B = 45°. Justify your steps and mention one real-life application of such constructions.
Answer:
Introduction

We studied constructing triangles using given angles and sides. Here, we use a ruler and protractor.


Argument 1
  • Draw AB = 5 cm.
  • At A, construct ∠A = 60° using a protractor.
  • At B, construct ∠B = 45°.
  • Extend arms to meet at C.

Argument 2

Our textbook shows similar constructions in Chapter 11. This method ensures accuracy.


Conclusion

Triangle ABC is constructed. Such constructions are used in designing roof trusses.

Question 9:
Construct a tangent to a circle of radius 4 cm from a point 8 cm away from its center. List steps and explain why right angles are crucial here.
Answer:
Introduction

We learned to draw tangents using the perpendicularity property. Here, we use a compass and ruler.


Argument 1
  • Draw circle with radius 4 cm (O as center).
  • Mark point P, 8 cm from O.
  • Join OP and bisect it to find midpoint M.
  • Draw semicircle on OP, intersecting circle at T.
  • PT is the tangent.

Argument 2

Our textbook shows ∠OTP = 90° ensures PT is tangent. Right angles guarantee perpendicularity.


Conclusion

Tangent PT is constructed. This is used in road curvature designs.

Question 10:
Construct a triangle ABC where BC = 7 cm, ∠B = 45°, and AB + AC = 13 cm. Justify your construction steps.
Answer:
Introduction

We studied how to construct triangles given specific conditions. Here, we need to draw ΔABC with BC = 7 cm, ∠B = 45°, and AB + AC = 13 cm.


Argument 1
  • Draw BC = 7 cm and construct ∠B = 45° using a protractor.
  • From point B, mark BD = 13 cm (AB + AC) on the ray.

Argument 2
  • Join D to C and draw the perpendicular bisector of DC.
  • The intersection of the bisector and BD gives point A.
  • Join A to C to complete ΔABC.

Conclusion

Our textbook shows this method ensures AB + AC = 13 cm. The construction is verified by measuring sides.

Question 11:
Construct a tangent to a circle of radius 4 cm from a point 8 cm away from its center. Write the steps and justification.
Answer:
Introduction

We learned to draw tangents to circles from external points. Here, we construct a tangent to a circle (radius = 4 cm) from a point 8 cm away.


Argument 1
  • Draw the circle with center O and radius 4 cm.
  • Mark point P, 8 cm from O, and join OP.

Argument 2
  • Construct the perpendicular bisector of OP to find midpoint M.
  • Draw a circle with radius OM and center M.
  • The intersection points (T, T') of both circles are the tangents.

Conclusion

Our textbook confirms PT and PT' are tangents as they are perpendicular to OT and OT'. The construction is verified using a protractor.

Question 12:
Construct a tangent to a circle of radius 4 cm from a point P 8 cm away from its center O. Explain each step with justification.
Answer:
Introduction

We learned in Chapter 10 how to draw tangents to a circle from an external point. Our NCERT textbook provides similar examples.


Argument 1
  • Draw a circle with radius 4 cm and mark its center O.
  • From O, locate point P at 8 cm and join OP.

Argument 2
  • Construct the perpendicular bisector of OP to find its midpoint M.
  • Draw a circle with radius OM and mark intersection points T1 and T2 with the original circle. PT1 and PT2 are the required tangents.

Conclusion

This method is applied in real-world problems like designing road curves or machinery parts.

Question 13:
Construct a triangle ABC where AB = 6 cm, ∠B = 45°, and BC = 5 cm. Justify your construction steps and verify the triangle using the SAS congruence rule.
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.
Step 2: At point B, use a protractor to construct an angle of 45° and mark it.
Step 3: From point B, measure 6 cm along the angle line to locate point A.
Step 4: Join points A and C to complete the triangle.

Verification using SAS rule: Since we have two sides (AB and BC) and the included angle (∠B) matching the given measurements, the triangle is correctly constructed as per the SAS congruence rule.

Question 14:
Construct a pair of tangents to a circle of radius 4 cm from a point 8 cm away from its center. Measure the lengths of the tangents and justify your construction steps.
Answer:

To construct the tangents, follow these steps:


Step 1: Draw a circle with radius 4 cm and mark its center as O.
Step 2: Mark a point P 8 cm away from O outside the circle.
Step 3: Join O and P to form the line segment OP.
Step 4: Draw the perpendicular bisector of OP to find its midpoint M.
Step 5: With M as the center and radius OM, draw a semicircle intersecting the original circle at points T and T'.
Step 6: Join PT and PT', which are the required tangents.

Measurement and Justification: The lengths of the tangents PT and PT' should be equal and can be calculated using the Pythagorean theorem:
√(OP² - OT²) = √(8² - 4²) = √(64 - 16) = √48 ≈ 6.93 cm.
This confirms the correctness of the construction.

Question 15:
Construct a triangle ABC in which BC = 7 cm, ∠B = 45°, and AB - AC = 3 cm. Justify your construction steps and verify the given conditions.
Answer:

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


Step 1: Draw the base BC = 7 cm.
Use a ruler to draw a line segment BC of length 7 cm.

Step 2: Construct ∠B = 45° at point B.
Using a protractor, draw an angle of 45° at point B.

Step 3: Mark point D on the ray from B such that BD = 3 cm.
This ensures AB - AC = 3 cm, as D will help locate A.

Step 4: Join D to C and draw the perpendicular bisector of DC.
The perpendicular bisector will intersect the ray from B at point A.

Step 5: Join A to C to complete the triangle ABC.

Justification:

  • Since A lies on the perpendicular bisector of DC, AD = AC.
  • Given BD = 3 cm, AB - AC = AB - AD = BD = 3 cm, satisfying the condition.
  • The angle at B is 45°, and BC = 7 cm, as required.

Verification: Measure AB and AC to confirm AB - AC = 3 cm and check ∠B = 45° using a protractor.

Question 16:
Construct a triangle ABC where AB = 6 cm, ∠B = 45°, and BC + CA = 8 cm. Justify each step of your construction and verify the given conditions.
Answer:

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


Step 1: Draw the base AB = 6 cm using a ruler.
Step 2: At point B, construct an angle of 45° using a protractor and mark it as ∠ABX.
Step 3: From point B, measure BD = 8 cm (since BC + CA = 8 cm) along BX.
Step 4: Join AD and draw its perpendicular bisector to intersect BD at point C.
Step 5: Join AC to complete the triangle.

Verification: Measure BC and CA using a ruler—their sum should be 8 cm. Also, confirm ∠B = 45° using a protractor.

Key Concept: The perpendicular bisector ensures CA = CD, so BC + CA = BC + CD = BD = 8 cm, satisfying the condition.

Question 17:
Construct a right-angled triangle where the hypotenuse is 5 cm and one of the legs is 3 cm. Explain the construction process and verify the properties.
Answer:

To construct the right-angled triangle with the given conditions:


Step 1: Draw the leg AB = 3 cm using a ruler.
Step 2: At point A, construct a 90° angle using a protractor and mark it as ∠BAX.
Step 3: With B as the center, draw an arc of radius 5 cm (hypotenuse) to intersect AX at point C.
Step 4: Join BC to complete the triangle.

Verification: Measure ∠A to confirm it is 90°. Use the Pythagorean theorem: 3² + AC² = 5² → AC = 4 cm (verify with a ruler).

Key Concept: The construction ensures the triangle satisfies the right-angle property and given side lengths. The hypotenuse is always opposite the right angle.

Question 18:
Construct a triangle ABC where AB = 5 cm, ∠B = 60°, and BC = 6 cm. Then, construct another triangle whose sides are 3/4 times the corresponding sides of ΔABC. Write the steps of construction clearly and justify your construction.
Answer:

To construct the required triangles, follow these steps:


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

Step 2: Construct a scaled-down triangle (3/4 ratio)
1. Below BC, draw a ray BX making an acute angle.
2. Mark 4 equal divisions (e.g., B1, B2, B3, B4) on BX using a compass.
3. Join B4 to C.
4. Draw a line parallel to B4C from B3 intersecting BC at C'.
5. From C', draw a line parallel to AC intersecting BA at A'.
6. ΔA'BC' is the required triangle with sides 3/4 of ΔABC.

Justification: By the Basic Proportionality Theorem, the sides are scaled proportionally, ensuring similarity.

Question 19:
Construct a tangent to a circle of radius 4 cm from a point 6 cm away from its center. Write the steps of construction and justify your answer with a diagram.
Answer:

To construct the tangent, follow these steps:


Step 1: Draw the circle and mark the external point
1. Draw a circle with center O and radius 4 cm.
2. Mark a point P at a distance of 6 cm from O.

Step 2: Construct the perpendicular bisector
1. Join O and P.
2. Find the midpoint M of OP using a compass.
3. With M as the center and radius OM, draw a semicircle intersecting the circle at T and T'.

Step 3: Draw the tangents
1. Join PT and PT', which are the required tangents.

Justification: Since ∠OTP = 90° (angle in a semicircle), PT is perpendicular to the radius OT, satisfying the tangent condition. The same applies to PT'.


Diagram: A circle with center O, point P outside, and two tangents PT and PT' touching the circle at T and T', respectively.

Question 20:
Construct a triangle ABC where AB = 5 cm, ∠B = 60°, and BC = 6 cm. Then, construct another triangle whose sides are 3/4 times the corresponding sides of ΔABC. Justify your construction steps.
Answer:

To construct ΔABC and its scaled-down version, follow these steps:


Step 1: Draw the base BC = 6 cm
Using a ruler, draw a line segment BC of length 6 cm.

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 AB = 5 cm
From point B, measure 5 cm along the ray and mark point A. Join A to C to complete ΔABC.

Step 4: Draw a downward ray from B
Construct an acute angle below BC and mark 4 equal arcs (B1, B2, B3, B4) using a compass.

Step 5: Join B4 to C
Draw a line from B4 to C. Now, draw a line parallel to B4C from B3, intersecting BC at C'.

Step 6: Construct the scaled triangle
From C', draw a line parallel to AC, intersecting AB at A'. ΔA'BC' is the required triangle with sides 3/4 of ΔABC.

Justification: The construction uses the Basic Proportionality Theorem, ensuring the sides of the new triangle are proportional to the original. The scaling factor 3/4 is achieved by dividing BB3/BB4.

Question 21:
Construct a tangent to a circle of radius 4 cm from a point 8 cm away from its center. Write the steps and justify your construction.
Answer:

To construct a tangent to the circle, follow these steps:


Step 1: Draw the circle and mark its center O
Using a compass, draw a circle with radius 4 cm and mark its center O.

Step 2: Mark the external point P
From O, measure 8 cm and mark point P outside the circle.

Step 3: Draw the perpendicular bisector of OP
Join O and P. Construct the perpendicular bisector of OP to find its midpoint M.

Step 4: Draw a semicircle with radius OM
Using M as the center and OM as radius, draw a semicircle intersecting the original circle at T.

Step 5: Join PT
PT is the required tangent to the circle.

Justification: The construction ensures that ∠OTP = 90° (by the Thales' theorem), making PT perpendicular to the radius OT. Hence, PT is a valid tangent as it touches the circle at exactly one point (T).

Question 22:
Construct a triangle ABC where AB = 6 cm, BC = 7 cm, and ∠B = 60°. Then, construct another triangle whose sides are 3/4 of the corresponding sides of ABC. Write the steps of construction clearly and justify your construction.
Answer:

To construct the required triangles, follow these steps:


Step 1: Construct ΔABC with given measurements
1. Draw a line segment AB = 6 cm.
2. At point B, construct an angle of 60° using a protractor.
3. From B, measure 7 cm and mark point C on the angle line.
4. Join A to C to complete ΔABC.

Step 2: Construct a similar triangle with sides 3/4 of ΔABC
1. Below AB, draw a ray AX at any acute angle.
2. Mark 4 equal points (A₁, A₂, A₃, A₄) on AX using a compass.
3. Join A₄ to B.
4. Draw a line parallel to A₄B from A₃ to intersect AB at B'.
5. From B', draw a line parallel to BC to intersect AC at C'.
6. ΔAB'C' is the required triangle with sides 3/4 of ΔABC.

Justification: By the Basic Proportionality Theorem, since B'C' is parallel to BC, the sides of ΔAB'C' are proportional to ΔABC. The ratio is 3:4 as constructed.

Question 23:
Construct a pair of tangents to a circle of radius 4 cm from a point 8 cm away from its center. Measure the lengths of the tangents and verify your construction mathematically.
Answer:

To construct the tangents and verify their lengths, follow these steps:


Step 1: Draw the circle and mark the external point
1. Draw a circle with center O and radius 4 cm.
2. Mark a point P 8 cm away from O.

Step 2: Construct the tangents
1. Join O and P.
2. Find the midpoint M of OP using a perpendicular bisector.
3. With M as the center and MO as radius, draw a circle intersecting the original circle at T and T'.
4. Join PT and PT', which are the required tangents.

Step 3: Measure and verify the lengths
1. Measure PT and PT' using a ruler. Both should be approximately 6.93 cm.
2. Mathematical Verification:
- By the tangent-secant theorem, PT = √(OP² - OT²).
- Substituting values: √(8² - 4²) = √(64 - 16) = √48 ≈ 6.93 cm.

Conclusion: The construction is correct as the measured and calculated lengths match.

Question 24:
Construct a triangle ABC where AB = 5 cm, ∠B = 60°, and BC = 6 cm. Then, construct another triangle whose sides are 4/3 times the corresponding sides of △ABC. Write the steps of construction clearly and justify your construction.
Answer:

To construct the required triangles, follow these steps:


Step 1: Construct △ABC
1. Draw a line segment BC = 6 cm.
2. At point B, construct an angle of 60° using a protractor.
3. From B, measure 5 cm along the angle line and mark point A.
4. Join A to C to complete △ABC.

Step 2: Construct a similar triangle with sides 4/3 times △ABC
1. Below BC, draw a ray BX making an acute angle.
2. Mark 3 equal divisions (B1, B2, B3) on BX using a compass.
3. Join B3 to C.
4. Draw a line parallel to B3C from B2 to meet BC extended at C'.
5. From C', draw a line parallel to AC to meet BA extended at A'.
6. △A'BC' is the required triangle.

Justification: By the Basic Proportionality Theorem, since the sides are scaled by 4/3, the new triangle is similar to the original with sides in the ratio 4:3.

Question 25:
Construct a tangent to a circle of radius 4 cm from a point 6 cm away from its center. Write the steps of construction and justify your answer.
Answer:

To construct the tangent, follow these steps:


Step 1: Draw the circle and mark the external point
1. Draw a circle with center O and radius 4 cm.
2. Mark a point P at a distance of 6 cm from O.

Step 2: Construct the perpendicular bisector
1. Join O and P.
2. Find the midpoint M of OP using a compass.
3. With M as the center and radius OM, draw a semicircle intersecting the circle at T.

Step 3: Draw the tangent
1. Join P to T.
2. PT is the required tangent.

Justification: Since ∠OTP is 90° (angle in a semicircle), PT is perpendicular to the radius OT. Hence, by the tangent property, PT is a valid tangent.

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 is asked to construct a triangle ABC where AB = 5 cm, ∠A = 60°, and ∠B = 45°. Explain the steps involved using ruler and compass.
Answer:
Problem Interpretation

We need to construct a triangle with given side and angles using ruler and compass.

Mathematical Modeling
  • Draw base AB = 5 cm.
  • At A, construct ∠A = 60° using protractor.
  • At B, construct ∠B = 45°.
Solution

The intersection point of the two rays is C. Join AC and BC. [Diagram: Triangle ABC with labeled sides and angles]

Question 2:
Construct a tangent to a circle of radius 4 cm from a point 6 cm away from its center. List the steps as per NCERT guidelines.
Answer:
Problem Interpretation

We studied constructing a tangent to a circle from an external point.

Mathematical Modeling
  • Draw circle with radius 4 cm and mark center O.
  • Mark point P, 6 cm from O.
  • Join OP and draw perpendicular bisector.
Solution

Bisect OP at M. Draw circle with radius OM. Intersection points are tangents. [Diagram: Circle with tangent lines from P]

Question 3:
A student needs to construct a triangle with sides 5 cm, 6 cm, and 7 cm. Explain the steps involved and justify why this construction is possible.
Answer:
Problem Interpretation

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


Mathematical Modeling
  • Check triangle inequality: 5 + 6 > 7, 5 + 7 > 6, 6 + 7 > 5.
  • Since all conditions hold, construction is possible.

Solution

Steps: (1) Draw base AB = 7 cm. (2) Draw arcs of radii 5 cm and 6 cm from A and B. (3) Intersection point C forms △ABC.

Question 4:
Construct a tangent to a circle of radius 3 cm from a point 7 cm away from its center. List the steps and verify the construction.
Answer:
Problem Interpretation

We studied how to draw a tangent to a circle from an external point.


Mathematical Modeling
  • Distance (7 cm) > radius (3 cm), so two tangents exist.
  • Use Pythagoras' theorem to find tangent length: √(7² - 3²) = √40 ≈ 6.32 cm.

Solution

Steps: (1) Draw circle and mark external point P. (2) Join OP. (3) Draw perpendicular bisector of OP. (4) Mark midpoint M. (5) Draw circle with radius OM. (6) Intersections give tangent points.

Question 5:
A student needs to construct a triangle ABC where AB = 5 cm, ∠A = 60°, and ∠B = 45°. Explain the steps using ruler and compass only.
Answer:
Problem Interpretation

We need to construct a triangle with given side and angles using ruler and compass.

Mathematical Modeling
  • Draw base AB = 5 cm.
  • At A, construct ∠A = 60° using protractor.
  • At B, construct ∠B = 45°.
Solution

The intersection point of the two angle lines is C. ABC is the required triangle. [Diagram: Triangle ABC with labeled sides and angles]

Question 6:
Construct a tangent to a circle of radius 3 cm from a point 5 cm away from its center. Justify your steps.
Answer:
Problem Interpretation

We studied how to draw a tangent to a circle from an external point.

Mathematical Modeling
  • Draw circle with radius 3 cm and mark center O.
  • Mark point P, 5 cm from O.
  • Join OP and bisect it to find midpoint M.
Solution

Draw semicircle on OP. Intersection with circle is tangent point T. PT is the required tangent. [Diagram: Circle with tangent PT]

Question 7:
Construct a tangent to a circle of radius 3 cm from a point 5 cm away from its center. Justify your steps.
Answer:
Problem Interpretation

We studied constructing tangents to a circle from an external point.

Mathematical Modeling
  • Draw circle with radius 3 cm and mark center O.
  • Mark point P 5 cm from O.
  • Join OP and draw perpendicular bisector.
Solution

Draw semicircle on OP. Intersection points with the circle are tangents. [Diagram: Circle with tangent lines from P] Our textbook shows this method ensures 90° angles.

Question 8:
A student needs to construct a triangle ABC where AB = 5 cm, BC = 6 cm, and AC = 7 cm. Explain the steps involved in the construction and justify why the triangle is unique.
Answer:
Problem Interpretation

We need to construct a triangle with given side lengths using a ruler and compass.

Mathematical Modeling
  • Draw base BC = 6 cm.
  • With B as center, draw an arc of radius 5 cm.
  • With C as center, draw an arc of radius 7 cm.
Solution

The arcs intersect at point A. Join A to B and C. The triangle is unique as all sides are fixed.

Question 9:
Construct a tangent to a circle of radius 3 cm from a point P located 7 cm away from its center O. List the steps and verify the construction.
Answer:
Problem Interpretation

We studied how to draw a tangent to a circle from an external point.

Mathematical Modeling
  • Draw circle with center O and radius 3 cm.
  • Mark point P 7 cm from O.
  • Bisect OP and draw a circle with diameter OP.
Solution

The two circles intersect at T. PT is the tangent. Verification: OT is perpendicular to PT.

Question 10:
A student needs to construct a triangle ABC where AB = 5 cm, ∠A = 60°, and ∠B = 45°. Explain the steps using ruler and compass and justify the construction.
Answer:
Problem Interpretation

We need to construct ΔABC with given sides and angles using ruler and compass.

Mathematical Modeling
  • Draw base AB = 5 cm.
  • At A, construct ∠A = 60° using protractor.
  • At B, construct ∠B = 45°.
Solution

The intersection of the two rays from A and B gives point C. We studied that the sum of angles in a triangle is 180°, so ∠C = 75°.

[Diagram: Triangle ABC with labeled sides and angles]
Question 11:
Construct a tangent to a circle of radius 4 cm from a point P located 8 cm away from its center O. List the steps and verify the construction.
Answer:
Problem Interpretation

We need to draw a tangent from external point P to a circle with center O and radius 4 cm.

Mathematical Modeling
  • Draw OP = 8 cm.
  • Bisect OP to find midpoint M.
  • Draw a circle with radius OM and center M.
Solution

The intersection of the two circles gives tangent points. Our textbook shows that the tangent is perpendicular to the radius at the point of contact, so ∠OTP = 90°.

[Diagram: Circle with tangent PT and radius OT]
Question 12:
A student is asked to construct a triangle ABC with sides AB = 5 cm, BC = 6 cm, and AC = 7 cm using a ruler and compass.

(i) Explain the steps involved in constructing the triangle. (2 marks)
(ii) Justify why the construction is possible with the given measurements. (2 marks)
Answer:

(i) Steps:
1. Draw base BC = 6 cm.
2. With B as center, draw an arc of radius 5 cm.
3. With C as center, draw an arc of radius 7 cm.
4. The intersection point of the arcs is A.
5. Join AB and AC to complete ΔABC.

(ii) Justification:
The construction is possible because the sum of any two sides (5 + 6 > 7, 5 + 7 > 6, 6 + 7 > 5) is greater than the third side, satisfying the triangle inequality.

Question 13:
A teacher asks students to construct a tangent to a circle of radius 3 cm from a point P located 8 cm away from its center O.

(i) Describe the construction steps. (2 marks)
(ii) Prove that the constructed line is a tangent. (2 marks)
Answer:

(i) Steps:
1. Draw circle with radius 3 cm and mark center O.
2. Join OP and bisect it to find midpoint M.
3. Draw a circle with radius OM and center M.
4. The intersection points of the two circles are tangents from P.

(ii) Proof:
The constructed line is perpendicular to the radius at the point of contact, as ∠OAP = 90° (by Thales' theorem). Hence, it is a tangent.

Question 14:
A construction problem involves dividing a 7 cm line segment AB in the ratio 3:2.

(i) Outline the steps to divide AB in the given ratio. (2 marks)
(ii) Verify your construction using the Basic Proportionality Theorem. (2 marks)
Answer:

(i) Steps:
1. Draw AB = 7 cm.
2. Draw a ray AX at any acute angle to AB.
3. Mark 5 points (3 + 2) on AX at equal distances.
4. Join the 5th point to B and draw a line parallel to it from the 3rd point to divide AB.

(ii) Verification:
By Basic Proportionality Theorem, the parallel line divides AB in the ratio 3:2, as the corresponding segments are proportional.

Question 15:

A farmer wants to construct a triangular field with sides 7 cm, 8 cm, and 9 cm using a compass and ruler. However, he only has a scale and protractor. Help him by describing the step-by-step construction process.

Answer:

Step-by-Step Construction:


1. Draw the base: Use the scale to draw a line segment AB of length 8 cm (the middle side for easier construction).


2. Mark arcs for sides: With A as the center, draw an arc of radius 7 cm using the protractor to measure the distance.


3. With B as the center, draw another arc of radius 9 cm intersecting the first arc at point C.


4. Complete the triangle: Join A to C and B to C to form △ABC.


Verification: Measure all sides with the scale to ensure accuracy (7 cm, 8 cm, 9 cm).

Question 16:

A student is asked to construct a tangent to a circle of radius 4 cm from a point 10 cm away from its center. Explain the construction process and justify each step.

Answer:

Construction Steps:


1. Draw the circle: With center O, draw a circle of radius 4 cm using a compass.


2. Mark the external point: Locate point P 10 cm away from O and join them.


3. Find the midpoint: Draw the perpendicular bisector of OP to find its midpoint M.


4. Draw the auxiliary circle: With M as the center and radius OM, draw a circle intersecting the original circle at T and T'.


5. Draw tangents: Join PT and PT', which are the required tangents.


Justification: Since ∠OTP is 90° (angle in a semicircle), PT is perpendicular to the radius, satisfying the tangent condition.

Question 17:

A farmer wants to construct a tangent to a circle of radius 4 cm from a point P outside the circle at a distance of 6 cm from its center. Help the farmer by providing step-by-step instructions for the construction.

Answer:

To construct a tangent from point P to the circle, follow these steps:


Step 1: Draw a circle with center O and radius 4 cm.
Step 2: Mark point P at a distance of 6 cm from O.
Step 3: Join OP and draw its perpendicular bisector to find its midpoint M.
Step 4: With M as the center and radius OM, draw a semicircle intersecting the original circle at T.
Step 5: Join PT, which is the required tangent.

Verification: Since ∠OTP = 90° (angle in a semicircle), PT is indeed a tangent.

Question 18:

Construct a triangle ABC with sides AB = 5 cm, BC = 6 cm, and AC = 7 cm. Then, construct another triangle whose sides are 4/5 times the corresponding sides of ΔABC. Explain the steps clearly.

Answer:

Follow these steps to construct the required triangle:


Step 1: Draw BC = 6 cm.
Step 2: With B as center and radius 5 cm, draw an arc.
Step 3: With C as center and radius 7 cm, draw another arc intersecting the first arc at A.
Step 4: Join AB and AC to complete ΔABC.
Step 5: Below BC, draw a ray BX making an acute angle.
Step 6: Mark 5 equal points on BX (B₁, B₂, ..., B₅).
Step 7: Join B₅C and draw a line parallel to it from B₄ to meet BC at C'.
Step 8: From C', draw a line parallel to AC to meet AB at A'.

ΔA'BC' is the required triangle with sides 4/5 of ΔABC.

Question 19:
A student is asked to construct a triangle ABC where AB = 5 cm, ∠A = 60°, and ∠B = 45°. However, the student mistakenly draws ∠B = 50° instead of 45°. Analyze the error and explain the correct steps to construct the triangle as per the given measurements.
Answer:

The student's error lies in incorrectly measuring ∠B as 50° instead of the required 45°. Here’s the correct step-by-step construction:


Step 1: Draw the base line segment AB = 5 cm.
Step 2: At point A, use a protractor to construct ∠A = 60°.
Step 3: At point B, construct ∠B = 45° (not 50°). The rays from A and B will intersect at point C.
Step 4: Join points A, B, and C to complete the triangle.

If ∠B is drawn as 50°, the triangle's shape and side lengths will differ, making it incorrect. Always verify angles with a protractor for accuracy.

Question 20:
A construction worker needs to divide a 7 cm long line segment AB into a ratio of 3:4 using a compass and ruler. Explain the geometric construction steps and justify why this method works.
Answer:

Dividing a line segment in a given ratio involves the concept of similar triangles. Here’s the correct procedure:


Step 1: Draw line segment AB = 7 cm.
Step 2: From point A, draw a ray AX at any acute angle.
Step 3: Using a compass, mark 7 equal divisions (since 3 + 4 = 7) on AX, naming them A₁, A₂, ..., A₇.
Step 4: Join A₇ to B.
Step 5: Draw a line parallel to A₇B from A₃ (since the ratio is 3:4) to intersect AB at point P.

AP:PB will now be 3:4. This method works because the parallel lines create similar triangles, ensuring proportional division of AB.

Question 21:
A student is asked to construct a triangle ABC where AB = 6 cm, ∠A = 60°, and ∠B = 45°. However, the student mistakenly draws ∠A = 45° and ∠B = 60°.

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

Answer:

The student's initial construction is incorrect because the angles and sides do not match the given conditions. Here's how to correct it:


Step 1: Draw the base AB = 6 cm using a ruler.
Step 2: At point A, use a protractor to construct ∠A = 60° (not 45°).
Step 3: At point B, construct ∠B = 45° (not 60°).
Step 4: The intersection of the two arms of the angles will give point C, completing the triangle.

Justification: The sum of angles in a triangle must be 180°. The given angles 60° + 45° = 105°, leaving ∠C = 75°, which is valid. The student's attempt violated the given conditions, leading to an incorrect triangle.

Question 22:
A gardener wants to divide a circular flower bed of radius 4 cm into six equal parts using only a compass and ruler.

Describe the step-by-step construction process and explain how this ensures equal division.

Answer:

To divide the circular flower bed into six equal parts, follow these steps:


Step 1: Draw a circle with radius 4 cm and mark its center O.
Step 2: Draw a diameter AB of the circle.
Step 3: With A as the center and the same radius, draw arcs intersecting the circle at points C and D.
Step 4: Repeat the process with B as the center, marking points E and F.
Step 5: Connect the points A, C, E, B, F, D to divide the circle into six equal sectors.

Explanation: Since the radius is constant, each arc subtends a central angle of 60° (360° ÷ 6). This ensures all six sectors are congruent, dividing the circle equally.

Question 23:
A student is asked to construct a triangle ABC where AB = 5 cm, BC = 6 cm, and ∠B = 60°. The student is also required to draw a circle circumscribing the triangle. Explain the steps involved in the construction and justify the method used.
Answer:

To construct triangle ABC and its circumcircle, follow these steps:


Step 1: Draw the base BC = 6 cm using a ruler.
Step 2: At point B, construct an angle of 60° using a protractor or compass. Mark point A such that AB = 5 cm.
Step 3: Join points A and C to complete the triangle ABC.
Step 4: To draw the circumcircle, find the perpendicular bisectors of any two sides (e.g., AB and BC).
Step 5: The point where the perpendicular bisectors intersect is the circumcenter. Use this point as the center and the distance to any vertex as the radius to draw the circumcircle.

Justification: The circumcircle passes through all three vertices of the triangle because the circumcenter is equidistant from them. This method ensures accuracy and adherence to geometric principles.

Question 24:
A teacher asks students to construct a pair of tangents to a circle of radius 4 cm from a point 8 cm away from its center. Describe the construction process and verify the length of the tangents.
Answer:

To construct the tangents and verify their length, follow these steps:


Step 1: Draw a circle with radius 4 cm and mark its center O.
Step 2: Mark a point P 8 cm away from O. Join OP.
Step 3: Find the midpoint M of OP and draw a circle with diameter OP. This circle intersects the original circle at points T and T′.
Step 4: Join PT and PT′, which are the required tangents.
Verification: Since ∠OTP = 90° (angle in a semicircle), PT is a tangent. Using Pythagoras' theorem in ΔOTP:
PT = √(OP² - OT²) = √(8² - 4²) = √(64 - 16) = √48 = 4√3 cm.

Note: Both tangents will be equal in length due to the tangent-segment theorem.

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