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

Understand the Question

Let $P$ be an external point from which two tangents $PA$ and $PB$ are drawn to a circle with centre $O$ , touching the circle at $A$ and $B$ .
The line segment joining the points of contact to the centre gives radii $OA$ and $OB$ .
The tangent at any point is perpendicular to the radius through the point of contact, making $\angle OAP = \angle OBP = 90^\circ$ .
In quadrilateral $OAPB$ , the sum of all four interior angles is $360^\circ$ . Using this property, we show that $\angle APB + \angle BOA = 180^\circ$ , which proves the two angles are supplementary.

Step 1 · Find Angles at Points of Contact

Let $PA$ and $PB$ be tangents drawn from an external point $P$ to a circle with centre $O$ , with points of contact at $A$ and $B$ respectively. Diagram 1

Since the radius is perpendicular to the tangent at the point of contact:

OA \perp PA \implies \angle OAP = 90^\circ

OB \perp PB \implies \angle OBP = 90^\circ

Step 2 · Apply Angle Sum Property of Quadrilateral

In quadrilateral $OAPB$ , the sum of all interior angles is $360^\circ$ :

\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

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

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

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

Answer

Hence proved that $\angle APB + \angle BOA = 180^\circ$ (supplementary).

Common Mistakes

Supplementary vs. Complementary: Confusing supplementary angles (sum $= 180^\circ$ ) with complementary angles (sum $= 90^\circ$ ).
Tangent-Radius Theorem: Forgetting that the angle between a radius and a tangent at the point of contact is strictly $90^\circ$ ( $OA \perp PA$ and $OB \perp PB$ ).

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 $24\text{ cm}$ and the distance of $Q$ from the centre is $25\text{ cm}$ . The radius of the circle is (A) $7\text{ cm}$ (B) $12\text{ cm}$ (C) $15\text{ cm}$ (D) $24.5\text{ cm}$

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 $5\text{ cm}$ from the centre of the circle is $4\text{ cm}$ . Find the radius of the circle.

Two concentric circles are of radii $5 \text{ cm}$ and $3 \text{ cm}$ . 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 $X'Y'$ are two parallel tangents to a circle with centre $O$ and another tangent $AB$ with point of contact $C$ intersecting $XY$ at $A$ and $X'Y'$ at $B$ . Prove that $\angle AOB = 90^\circ$ .

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 $4 \text{ cm}$ such that the segments $BD$ and $DC$ into which $BC$ is divided by the point of contact $D$ are of lengths $8 \text{ cm}$ and $6 \text{ cm}$ 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.