Circles | Exercise 10.2

Question 7

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.

Check your answer with HomiSolve it yourself, then let Homi check your steps and spot mistakes.
Solution

Concentric Circles: Two or more circles that share the same centre but have different radii. Here, both circles are centred at O with radii 3 cm and 5 cm.

Chord: A line segment whose both endpoints lie on a circle. Here, PQ is a chord of the larger circle that also touches the smaller circle.

Step 1 — Set up the geometric representation

Let OO be the common center of the two concentric circles. Let PQPQ be the chord of the larger circle that touches the smaller circle at point AA. Therefore, PQPQ is a tangent to the smaller circle at AA. The radius of the smaller circle is OA=3 cmOA = 3\text{ cm}. The radius of the larger circle is OP=5 cmOP = 5\text{ cm}. Tangent-Radius Perpendicularity Theorem: The radius to the point of tangency is always perpendicular to the tangent. So OA ⊥ PQ.

Since the tangent is perpendicular to the radius at the point of contact: OA⊥PQOA \perp PQ Thus, ∠OAP=90∘\angle OAP = 90^\circ. Therefore, △OAP\triangle OAP is a right-angled triangle.

Diagram 1

Step 2 — Calculate the length of APAP

Pythagoras Theorem: In a right-angled triangle, hypotenuse² = sum of squares of the other two sides. Here OP is the hypotenuse.

Apply the Pythagoras theorem in the right-angled triangle OAPOAP: OA2+AP2=OP2OA^2 + AP^2 = OP^2

Substitute the values of OAOA and OPOP: 32+AP2=523^2 + AP^2 = 5^2

Calculate the squares: 9+AP2=259 + AP^2 = 25

Subtract 99 from both sides: AP2=25−9AP^2 = 25 - 9

AP2=16AP^2 = 16

Take the square root: AP=16AP = \sqrt{16}

AP=4 cmAP = 4\text{ cm}

Step 3 — Find the total length of the chord PQPQ

Perpendicular Bisector Theorem (Chord): The perpendicular drawn from the centre of a circle to any chord always bisects that chord — it divides it into two equal halves.

The perpendicular from the center of a circle to a chord bisects the chord. Since OA⊥PQOA \perp PQ, AA is the midpoint of PQPQ. Therefore, the length of the chord is twice of APAP: PQ=2×APPQ = 2 \times AP

Substitute the value of APAP: PQ=2×4 cmPQ = 2 \times 4\text{ cm}

PQ=8 cmPQ = 8\text{ cm}

Answer

The length of the chord of the larger circle is 8 cm.

More questions in Exercise 10.2

Q1

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

  1. 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
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 A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.

Q7

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.

Q8

A quadrilateral ABCD 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, XY and X′Y′\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 X′Y′\text{X}'\text{Y}' at B. Prove that ∠AOB=90∘\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.

← Back to Circles