Circles | Exercise 10.2

Question 7

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.

Check your answer with HomiSolve it yourself, then let Homi check your steps and spot mistakes.
Solution
Understand the Question
  • Concentric Circles: Circles that share the same centre OO but have different radii (3 cm3\text{ cm} and 5 cm5\text{ cm}).
  • A chord PQPQ of the larger circle touches the smaller circle at point AA, making PQPQ a tangent to the smaller circle at AA.
  • By the tangent-radius theorem, the radius OAOA is perpendicular to PQPQ (OA⊥PQOA \perp PQ), forming a right-angled triangle △OAP\triangle OAP.
  • We use the Pythagoras theorem in △OAP\triangle OAP to find the length of APAP.
  • Since the perpendicular from the centre to a chord bisects the chord, the total chord length is PQ=2×APPQ = 2 \times AP.

Step 1 · Find the Length of APAP

Let OO be the common centre of the two concentric circles. Let PQPQ be the chord of the larger circle touching the smaller circle at point AA.Diagram 1

Radius of smaller circle, OA=3 cmOA = 3\text{ cm} Radius of larger circle, OP=5 cmOP = 5\text{ cm}

Since the tangent is perpendicular to the radius at the point of contact: OA⊥PQ  ⟹  ∠OAP=90∘OA \perp PQ \implies \angle OAP = 90^\circ

Applying Pythagoras theorem in right △OAP\triangle OAP:

OA2+AP2=OP232+AP2=529+AP2=25AP2=25−9=16AP=16=4 cm\begin{aligned} OA^2 + AP^2 &= OP^2 \\[0.6em] 3^2 + AP^2 &= 5^2 \\[0.6em] 9 + AP^2 &= 25 \\[0.6em] AP^2 &= 25 - 9 = 16 \\[0.6em] AP &= \sqrt{16} = 4\text{ cm} \end{aligned}

Step 2 · Find the Total Length of Chord PQPQ

The perpendicular drawn from the centre of a circle to a chord bisects the chord.

Since OA⊥PQOA \perp PQ, AA is the midpoint of PQPQ:

PQ=2×AP=2×4 cm=8 cm\begin{aligned} PQ &= 2 \times AP \\[0.6em] &= 2 \times 4\text{ cm} \\[0.6em] &= 8\text{ cm} \end{aligned}
Answer

8 cm8\text{ cm}

Common Mistakes
  • Stopping at Half-Length: Forgetting to double APAP, mistakenly reporting 4 cm4\text{ cm} instead of the full chord length PQ=8 cmPQ = 8\text{ cm}.
  • Hypotenuse Confusion: Confusing the radii — the radius of the larger circle (OP=5 cmOP = 5\text{ cm}) is the hypotenuse, while the radius of the smaller circle (OA=3 cmOA = 3\text{ cm}) is one of the legs.

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.

← Back to Circles