Circles | Exercise 10.2

Question 10

Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.

Solution

Supplementary Angles: Two angles are supplementary if their sum is 180°. This proof shows that ∠APB + ∠BOA = 180°, making them supplementary.

Angle subtended at the centre: ∠BOA is the angle formed at the centre O by the two radii OA and OB — the radii drawn to the points of contact of the tangents.

Let's prove the relationship between the angles.

Step 1 — Identify perpendicular radii

Let O be the center of the circle. Let P be an external point. PA and PB are tangents from P. A and B are points of contact. OA is the radius to tangent PA. OB is the radius to tangent PB. Tangent-Radius Perpendicularity Theorem: The radius drawn to the point of tangency is always perpendicular to the tangent.

A radius is perpendicular to the tangent. So, angle OAP is 90°. Angle OBP is also 90°.

$\angle OAP = 90^\circ$

$\angle OBP = 90^\circ$

Diagram 1

Step 2 — Sum of angles in quadrilateral

Consider the quadrilateral OAPB.

Angle Sum of a Quadrilateral: The four interior angles of any quadrilateral always add up to 360°.

The sum of angles in a quadrilateral is 360°. Let's add all the angles.

$\angle OAP + \angle APB + \angle OBP + \angle BOA = 360^\circ$

Substitute the known angle values.

$90^\circ + \angle APB + 90^\circ + \angle BOA = 360^\circ$

Combine the constant terms.

$180^\circ + \angle APB + \angle BOA = 360^\circ$

Subtract 180° from both sides.

$\angle APB + \angle BOA = 360^\circ - 180^\circ$

$\boxed{\angle APB + \angle BOA = 180^\circ}$

Answer

The angle between the two tangents ( $\angle APB$ ) and the angle subtended by the line-segment joining the points of contact at the centre ( $\angle BOA$ ) are supplementary.

More questions in Exercise 10.2

In Q.1 to 3, choose the correct option and give justification.

From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is (A) 7 cm (B) 12 cm (C) 15 cm (D) 24.5 cm

In Q.1 to 3, choose the correct option and give justification.

In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so that $\angle \text{POQ} = 110^\circ$ , then $\angle \text{PTQ}$ is equal to (A) $60^\circ$ (B) $70^\circ$ (C) $80^\circ$ (D) $90^\circ$

In Q.1 to 3, choose the correct option and give justification.

If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of $80^\circ$ , then $\angle \text{POA}$ is equal to (A) $50^\circ$ (B) $60^\circ$ (C) $70^\circ$ (D) $80^\circ$

Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.

The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.

Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

A quadrilateral ABCD is drawn to circumscribe a circle (see Fig. 10.12). Prove that

$\text{AB} + \text{CD} = \text{AD} + \text{BC}$

In Fig. 10.13, XY and $\text{X}'\text{Y}'$ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and $\text{X}'\text{Y}'$ at B. Prove that $\angle\text{AOB} = 90^\circ$ .

Q10

Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.

Q11

Prove that the parallelogram circumscribing a circle is a rhombus.

Q12

A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively (see Fig. 10.14). Find the sides AB and AC.

Q13

Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.