Circles | Exercise 10.2

Question 5

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

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Solution
Understand the Question
  • We need to prove that the line drawn perpendicular to a tangent at its point of contact passes through the centre OO of the circle.
  • We use the method of proof by contradiction:
    • Assume the perpendicular line does not pass through the centre OO, but through another point O′O'.
    • Compare this with the established theorem that the radius drawn to the point of contact is perpendicular to the tangent (OP⊥ABOP \perp AB, so ∠OPB=90∘\angle OPB = 90^\circ).
    • Show that both angles being 90∘90^\circ is only possible if the two lines coincide.

Step 1 · Assume the perpendicular passes through another point

Let a circle have centre OO and tangent ABAB touching the circle at point PP.Suppose the perpendicular to ABAB at PP does not pass through the centre OO. Let it pass through another point O′O'.

Therefore, ∠O′PB=90∘…(1)\angle O'PB = 90^\circ \quad \dots (1)

Step 2 · Apply the tangent-radius perpendicularity theorem

The radius drawn to the point of contact is perpendicular to the tangent: OP⊥ABOP \perp AB

Therefore, ∠OPB=90∘…(2)\angle OPB = 90^\circ \quad \dots (2)

Step 3 · Establish the contradiction

Comparing equations (1)(1) and (2)(2): ∠O′PB=∠OPB=90∘\angle O'PB = \angle OPB = 90^\circ

From the figure, a part cannot be equal to the whole. This equality is possible only if the line segment O′PO'P coincides with OPOP.

Therefore, our assumption is false, and the perpendicular at the point of contact to the tangent must pass through the centre OO.

Answer

Hence proved, the perpendicular at the point of contact to the tangent to a circle passes through the centre.

Common Mistakes
  • Skipping the Contradiction Argument: Simply stating that OP⊥ABOP \perp AB without showing why another perpendicular O′PO'P is impossible.
  • Missing Axiom: Not mentioning that a part cannot equal the whole, which is the geometric basis that forces O′PO'P and OPOP to coincide.

More questions in Exercise 10.2

Q1

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

  1. From a point QQ, the length of the tangent to a circle is 24 cm24\text{ cm} and the distance of QQ from the centre is 25 cm25\text{ cm}. The radius of the circle is (A) 7 cm7\text{ cm} (B) 12 cm12\text{ cm} (C) 15 cm15\text{ cm} (D) 24.5 cm24.5\text{ cm}
Q2

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

  1. In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so that ∠POQ=110∘\angle \text{POQ} = 110^\circ, then ∠PTQ\angle \text{PTQ} is equal to (A) 60∘60^\circ (B) 70∘70^\circ (C) 80∘80^\circ (D) 90∘90^\circ
Q3

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

  1. If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80∘80^\circ, then ∠POA\angle \text{POA} is equal to (A) 50∘50^\circ (B) 60∘60^\circ (C) 70∘70^\circ (D) 80∘80^\circ
Q4

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

Q5

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

Q6

The length of a tangent from a point AA at distance 5 cm5\text{ cm} from the centre of the circle is 4 cm4\text{ cm}. Find the radius of the circle.

Q7

Two concentric circles are of radii 5 cm5 \text{ cm} and 3 cm3 \text{ cm}. Find the length of the chord of the larger circle which touches the smaller circle.

Q8

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

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

Q9

In Fig. 10.13, XYXY and X′Y′X'Y' are two parallel tangents to a circle with centre OO and another tangent ABAB with point of contact CC intersecting XYXY at AA and X′Y′X'Y' at BB. Prove that ∠AOB=90∘\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 ABCABC is drawn to circumscribe a circle of radius 4 cm4 \text{ cm} such that the segments BDBD and DCDC into which BCBC is divided by the point of contact DD are of lengths 8 cm8 \text{ cm} and 6 cm6 \text{ cm} respectively (see Fig. 10.14). Find the sides ABAB and ACAC.

Q13

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

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