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

Supplementary angles add up to $180^\circ$ . We need to prove that $\angle \text{APB} + \angle \text{BOA} = 180^\circ$ .
The radius drawn to the point of contact is perpendicular to the tangent, giving two right angles: $\angle \text{OAP} = 90^\circ$ and $\angle \text{OBP} = 90^\circ$ .
In the four-sided figure (quadrilateral) $\text{OAPB}$ , the sum of all interior angles is $360^\circ$ .

Step 1 · Identify Perpendicular Radii

Let $O$ be the centre of the circle, $P$ be an external point, and $PA, PB$ be the two tangents touching the circle at points $A$ and $B$ . Diagram 1

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

\angle \text{OAP} = 90^\circ, \quad \angle \text{OBP} = 90^\circ

Step 2 · Apply Angle Sum Property of Quadrilateral

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

\angle \text{OAP} + \angle \text{APB} + \angle \text{OBP} + \angle \text{BOA} = 360^\circ

Substitute $\angle \text{OAP} = 90^\circ$ and $\angle \text{OBP} = 90^\circ$ :

\begin{aligned} 90^\circ + \angle \text{APB} + 90^\circ + \angle \text{BOA} &= 360^\circ \\ 180^\circ + \angle \text{APB} + \angle \text{BOA} &= 360^\circ \\ \angle \text{APB} + \angle \text{BOA} &= 360^\circ - 180^\circ \\ \angle \text{APB} + \angle \text{BOA} &= 180^\circ \end{aligned}

Answer

$\angle \text{APB} + \angle \text{BOA} = 180^\circ$ (Hence proved, the angles are supplementary)

Common Mistakes

Supplementary vs. Complementary: Confusing supplementary angles (sum $= 180^\circ$ ) with complementary angles (sum $= 90^\circ$ ).
Tangent-Radius Theorem: Forgetting that a tangent is strictly perpendicular to the radius at the point of contact ( $\angle \text{OAP} = \angle \text{OBP} = 90^\circ$ ).

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.