Madhyamik Mathematics Suggestion – Construction of Tangents to a Circle | Chapter 17

The chapter Construction of Tangents to a Circle is an important part of Madhyamik Mathematics. It is especially useful for construction-based and proof-based questions.

Students should practise the constructions carefully and also understand the geometric reasoning behind the figures. Knowledge of tangents, radii, chords, diameters and angle relationships is essential for solving the questions correctly.

Construction of Tangents to a Circle

In this chapter, students are expected to work with circles and tangents through both geometrical construction and mathematical proof. The important questions include proving relationships between angles and lines as well as constructing circles, triangles and tangents using given measurements.

Question	Main Topic	Type	Marks
1	Circle, tangents and diameter	Proof	5
2	Chord, tangent and central angle	Proof	5
3	Circle with a given diameter and tangents	Construction	5
4	Circumcircle and tangents of an equilateral triangle	Construction	5
5	Equilateral triangle circumscribed about a circle	Construction	5

Important Proof-Based Questions

Question 1: Prove that OA ∥ RQ

A circle has centre O. From an external point A, two tangents AP and AQ are drawn to the circle, where P and Q are the points of contact. PR is a diameter of the circle. Prove that:

OA ∥ RQ

Figure Placeholder — Question 1 Circle with centre O, external point A, tangents AP and AQ, points of contact P and Q, and diameter PR.
Use the original source figure here if an exact textbook-style figure is required.

Solution

Since AP and AQ are tangents drawn from the same external point A, the tangent segments are equal.

AP = AQ

Also, OP and OQ are radii of the same circle. Therefore:

OP = OQ

Consider the angle relationships produced by the two tangents and the radii. Let the relevant angle at the point of contact be x. The angle subtended by the corresponding arc at the centre is twice the angle subtended at the circumference.

∠POQ = 2x

Since PR is a diameter, the points P, O, R are collinear. Using the resulting angle relationship in triangle ORQ, we obtain:

∠ORQ = x

The corresponding angle made by OA with OR is supplementary to this angle. Hence:

∠ORQ + ∠ROA = 180°

Therefore:

OA ∥ RQ

Hence proved.

Question 2: Prove that ∠AOB = 2∠PAB

In a circle with centre O, AB is a chord and AP is a tangent to the circle at A. Prove that:

∠AOB = 2∠PAB

Figure Placeholder — Question 2 Circle with centre O, chord AB and tangent AP at A.
Use the original source figure here if an exact textbook-style figure is required.

Solution

Let:

∠AOB = 2x

Since OA and OB are radii of the same circle:

OA = OB

Therefore, triangle AOB is an isosceles triangle. Hence its two base angles are equal.

Since AP is tangent to the circle at A, the radius drawn to the point of contact is perpendicular to the tangent.

OA ⟂ AP

Therefore:

∠OAB + ∠PAB = 90°

From the isosceles triangle and the above right-angle relationship, we obtain:

∠PAB = x

But:

∠AOB = 2x

Therefore:

∠AOB = 2∠PAB

Hence proved.

Construction-Based Questions

Question 3: Construct a Circle with XY as Diameter and Draw Tangents

Draw a line segment XY of length 8 cm. Construct a circle taking XY as its diameter. Draw tangents to the circle at X and Y. State the relationship between the two tangents.

Figure 1: Circle with XY as diameter and tangents at X and Y

Construction Guidance

Draw the line segment XY = 8 cm.
Bisect XY and mark its midpoint as O.
With O as centre and OX as radius, draw the circle passing through X and Y.
Draw a line through X perpendicular to OX. This is the tangent at X.
Draw a line through Y perpendicular to OY. This is the tangent at Y.

Both tangents are perpendicular to the same straight line XY.

Therefore, the two tangents are parallel.

Question 4: Construct an Equilateral Triangle and Its Circumcircle

Construct an equilateral triangle ABC with side 5 cm. Construct its circumcircle. Then draw tangents to the circumcircle at A, B and C.

Figure 2: Equilateral triangle, circumcircle and tangents at A, B and C

Construction Guidance

Draw AB = 5 cm.
With A as centre and radius 5 cm, draw an arc.
With B as centre and radius 5 cm, draw another arc cutting the first arc at C.
Join AC and BC. The resulting triangle ABC is equilateral.
Construct the perpendicular bisectors of two sides of the triangle. Their point of intersection is the circumcentre O.
With O as centre and OA as radius, draw the circumcircle passing through A, B and C.
Draw a line through A perpendicular to OA. This is the tangent at A.
Similarly, draw lines through B and C perpendicular to OB and OC respectively.

Question 5: Construct an Equilateral Triangle Circumscribed About a Circle

Construct an equilateral triangle circumscribed about a circle whose radius is 3 cm.

Figure 3: Equilateral triangle circumscribed about a circle of radius 3 cm

Construction Guidance

Draw a circle with centre O and radius 3 cm.
Construct three radii so that the three central angles are equal.
Since a complete angle is 360°, each central angle is 120°.
Let the three radii meet the circle at three points.
Draw tangents to the circle at those three points. Each tangent is perpendicular to the corresponding radius.
Extend the three tangents until they intersect pairwise.
The three intersection points form the required equilateral triangle.

The given circle lies inside the triangle and touches all three sides. Therefore, it is the incircle of the constructed equilateral triangle.

Important Properties to Remember

Concept	Important Property
Radius and Tangent	The radius drawn to the point of contact is perpendicular to the tangent.
Two Tangents	Tangents drawn from the same external point to a circle are equal.
Radii	All radii of a circle are equal.
Diameter	A diameter passes through the centre of the circle.
Tangents at Diameter Endpoints	The tangents drawn at the two endpoints of a diameter are parallel.
Central Angle	The angle subtended by an arc at the centre is twice the angle subtended by the same arc at the circumference.

Preparation Tips for Madhyamik Examination

Practise every construction repeatedly so that the sequence of steps becomes familiar.
Use a sharp pencil, ruler and compass to keep the construction accurate.
Remember the relationship between a tangent and the radius at the point of contact.
Learn the equality of tangents drawn from the same external point.
In proof questions, write the reason behind each important step instead of writing only the final result.
Keep the construction figure clean and label every required point clearly.
Practise both construction and proof questions because the chapter requires understanding of the figure as well as the construction method.

Important Questions at a Glance

No.	Question	Focus Area
1	Prove that OA is parallel to RQ when PR is a diameter.	Geometrical Proof
2	Prove that ∠AOB = 2∠PAB.	Geometrical Proof
3	Construct a circle with an 8 cm diameter and draw tangents at the endpoints.	Construction
4	Construct the circumcircle of an equilateral triangle of side 5 cm and draw tangents at its vertices.	Construction
5	Construct an equilateral triangle circumscribed about a circle of radius 3 cm.	Construction

Important Note

The source material provides the five important questions for this chapter and solution material for the first two proof-based questions. It does not provide detailed written construction procedures for Questions 3, 4 and 5. The diagrams for Questions 3, 4 and 5 above are clean educational schematics added to make the construction concepts easier to understand.

The figures for Questions 1 and 2 are intentionally marked as placeholders because the source solution figures are specific textbook-style diagrams. The original source figures can be inserted in those locations when exact source figures are required.

Related Madhyamik Mathematics Preparation

Students preparing for Madhyamik Mathematics can also explore the Madhyamik Suggestion 2027 All Subjects resource for broader subject-wise preparation.

Quick Revision Keywords

Circle Tangent Point of Contact Radius Diameter Chord Circumcircle Incircle Construction Geometrical Proof

The chapter Construction of Tangents to a Circle is an important part of Madhyamik Mathematics. It is especially useful for construction-based and proof-based questions.

Construction of Tangents to a Circle

Question	Main Topic	Type	Marks
1	Circle, tangents and diameter	Proof	5
2	Chord, tangent and central angle	Proof	5
3	Circle with a given diameter and tangents	Construction	5
4	Circumcircle and tangents of an equilateral triangle	Construction	5
5	Equilateral triangle circumscribed about a circle	Construction	5

Important Proof-Based Questions

Question 1: Prove that OA ∥ RQ

A circle has centre O. From an external point A, two tangents AP and AQ are drawn to the circle, where P and Q are the points of contact. PR is a diameter of the circle. Prove that:

OA ∥ RQ

Solution

Since AP and AQ are tangents drawn from the same external point A, the tangent segments are equal.

AP = AQ

Also, OP and OQ are radii of the same circle. Therefore:

OP = OQ

∠POQ = 2x

Since PR is a diameter, the points P, O, R are collinear. Using the resulting angle relationship in triangle ORQ, we obtain:

∠ORQ = x

The corresponding angle made by OA with OR is supplementary to this angle. Hence:

∠ORQ + ∠ROA = 180°

Therefore:

OA ∥ RQ

Hence proved.

Question 2: Prove that ∠AOB = 2∠PAB

In a circle with centre O, AB is a chord and AP is a tangent to the circle at A. Prove that:

∠AOB = 2∠PAB

Figure Placeholder — Question 2 Circle with centre O, chord AB and tangent AP at A.
Use the original source figure here if an exact textbook-style figure is required.

Solution

Let:

∠AOB = 2x

Since OA and OB are radii of the same circle:

OA = OB

Therefore, triangle AOB is an isosceles triangle. Hence its two base angles are equal.

Since AP is tangent to the circle at A, the radius drawn to the point of contact is perpendicular to the tangent.

OA ⟂ AP

Therefore:

∠OAB + ∠PAB = 90°

From the isosceles triangle and the above right-angle relationship, we obtain:

∠PAB = x

But:

∠AOB = 2x

Therefore:

∠AOB = 2∠PAB

Hence proved.

Construction-Based Questions

Question 3: Construct a Circle with XY as Diameter and Draw Tangents

Draw a line segment XY of length 8 cm. Construct a circle taking XY as its diameter. Draw tangents to the circle at X and Y. State the relationship between the two tangents.

Figure 1: Circle with XY as diameter and tangents at X and Y

Construction Guidance

Draw the line segment XY = 8 cm.
Bisect XY and mark its midpoint as O.
With O as centre and OX as radius, draw the circle passing through X and Y.
Draw a line through X perpendicular to OX. This is the tangent at X.
Draw a line through Y perpendicular to OY. This is the tangent at Y.

Both tangents are perpendicular to the same straight line XY.

Therefore, the two tangents are parallel.

Question 4: Construct an Equilateral Triangle and Its Circumcircle

Construct an equilateral triangle ABC with side 5 cm. Construct its circumcircle. Then draw tangents to the circumcircle at A, B and C.

Figure 2: Equilateral triangle, circumcircle and tangents at A, B and C

Construction Guidance

Draw AB = 5 cm.
With A as centre and radius 5 cm, draw an arc.
With B as centre and radius 5 cm, draw another arc cutting the first arc at C.
Join AC and BC. The resulting triangle ABC is equilateral.
Construct the perpendicular bisectors of two sides of the triangle. Their point of intersection is the circumcentre O.
With O as centre and OA as radius, draw the circumcircle passing through A, B and C.
Draw a line through A perpendicular to OA. This is the tangent at A.
Similarly, draw lines through B and C perpendicular to OB and OC respectively.

Question 5: Construct an Equilateral Triangle Circumscribed About a Circle

Construct an equilateral triangle circumscribed about a circle whose radius is 3 cm.

Figure 3: Equilateral triangle circumscribed about a circle of radius 3 cm

Construction Guidance

Draw a circle with centre O and radius 3 cm.
Construct three radii so that the three central angles are equal.
Since a complete angle is 360°, each central angle is 120°.
Let the three radii meet the circle at three points.
Draw tangents to the circle at those three points. Each tangent is perpendicular to the corresponding radius.
Extend the three tangents until they intersect pairwise.
The three intersection points form the required equilateral triangle.

The given circle lies inside the triangle and touches all three sides. Therefore, it is the incircle of the constructed equilateral triangle.

Important Properties to Remember

Concept	Important Property
Radius and Tangent	The radius drawn to the point of contact is perpendicular to the tangent.
Two Tangents	Tangents drawn from the same external point to a circle are equal.
Radii	All radii of a circle are equal.
Diameter	A diameter passes through the centre of the circle.
Tangents at Diameter Endpoints	The tangents drawn at the two endpoints of a diameter are parallel.
Central Angle	The angle subtended by an arc at the centre is twice the angle subtended by the same arc at the circumference.

Preparation Tips for Madhyamik Examination

Practise every construction repeatedly so that the sequence of steps becomes familiar.
Use a sharp pencil, ruler and compass to keep the construction accurate.
Remember the relationship between a tangent and the radius at the point of contact.
Learn the equality of tangents drawn from the same external point.
In proof questions, write the reason behind each important step instead of writing only the final result.
Keep the construction figure clean and label every required point clearly.
Practise both construction and proof questions because the chapter requires understanding of the figure as well as the construction method.

Important Questions at a Glance

No.	Question	Focus Area
1	Prove that OA is parallel to RQ when PR is a diameter.	Geometrical Proof
2	Prove that ∠AOB = 2∠PAB.	Geometrical Proof
3	Construct a circle with an 8 cm diameter and draw tangents at the endpoints.	Construction
4	Construct the circumcircle of an equilateral triangle of side 5 cm and draw tangents at its vertices.	Construction
5	Construct an equilateral triangle circumscribed about a circle of radius 3 cm.	Construction

Important Note

Related Madhyamik Mathematics Preparation

Students preparing for Madhyamik Mathematics can also explore the Madhyamik Suggestion 2027 All Subjects resource for broader subject-wise preparation.

Quick Revision Keywords

Circle Tangent Point of Contact Radius Diameter Chord Circumcircle Incircle Construction Geometrical Proof

Madhyamik Mathematics Suggestion – Construction of Tangents to a Circle | Chapter 17

Construction of Tangents to a Circle

Important Proof-Based Questions

Solution

Solution

Construction-Based Questions

Construction Guidance

Construction Guidance

Construction Guidance

Important Properties to Remember

Preparation Tips for Madhyamik Examination

Important Questions at a Glance

Important Note

Related Madhyamik Mathematics Preparation

Quick Revision Keywords

Frequently Asked Questions

Madhyamik Mathematics Suggestion – Construction of Tangents to a Circle | Chapter 17

Construction of Tangents to a Circle

Important Proof-Based Questions

Solution

Solution

Construction-Based Questions

Construction Guidance

Construction Guidance

Construction Guidance

Important Properties to Remember

Preparation Tips for Madhyamik Examination

Important Questions at a Glance

Important Note

Related Madhyamik Mathematics Preparation

Quick Revision Keywords

Frequently Asked Questions