Circles — Practice

Question 1

How many tangents can be drawn to a circle from a point outside it?

Accepted Answer

Step 1 — From an external point exactly two tangents can be drawn.

✦ Answer: two.

Question 2

A tangent from a point 13 cm from the centre touches a circle of radius 5 cm. Find the tangent length.

Accepted Answer

Step 1 — AP = √(13² − 5²) = √(169 − 25) = √144.

✦ Answer: 12 cm.

Question 3

The radius OP meets tangent XY at P. What is ∠OPY?

Accepted Answer

Step 1 — The tangent is perpendicular to the radius at the contact point.

✦ Answer: 90°.

Question 4

From an external point A, tangents AP and AQ are drawn. If ∠PAQ = 50°, find ∠OPA and ∠POQ.

Accepted Answer

Step 1 — ∠OPA = 90° (tangent ⟂ radius).

Step 2 — In quadrilateral OPAQ, angles sum to 360°: ∠POQ = 360 − 90 − 90 − 50 = 130°.

✦ Answer: ∠OPA = 90°, ∠POQ = 130°.

Question 5

Two concentric circles have radii 5 cm and 3 cm. Find the length of the chord of the larger circle that touches the smaller.

Accepted Answer

Step 1 — The chord is tangent to the inner circle; the 3 cm radius is perpendicular and bisects it.

Step 2 — Half-chord = √(5² − 3²) = 4 cm ⇒ chord = 8 cm.

✦ Answer: 8 cm.

Question 6

Prove that the tangents drawn from an external point to a circle are equal in length.

Accepted Answer

Step 1 — Let tangents from A touch the circle (centre O) at P and Q. Join OP, OQ, OA.

Step 2 — In △OPA and △OQA: OP = OQ (radii), ∠OPA = ∠OQA = 90° (tangent ⟂ radius), OA common.

Step 3 — By RHS congruence, △OPA ≅ △OQA ⇒ AP = AQ. ∎

✦ Answer: AP = AQ by RHS congruence.

Question 7

Prove that a parallelogram circumscribing a circle is a rhombus.

Accepted Answer

Step 1 — For a circumscribing quadrilateral, AB + CD = BC + DA (equal tangents).

Step 2 — In a parallelogram AB = CD and BC = DA, so 2AB = 2BC ⇒ AB = BC.

Step 3 — Adjacent sides equal ⇒ all sides equal ⇒ rhombus. ∎

✦ Answer: it is a rhombus.

Question 8

A quadrilateral ABCD circumscribes a circle. If AB = 6 cm, BC = 7 cm and CD = 4 cm, find DA.

Accepted Answer

Step 1 — Opposite sides: AB + CD = BC + DA.

Step 2 — 6 + 4 = 7 + DA ⇒ DA = 3 cm.

✦ Answer: DA = 3 cm.

Question 9

A circle touches the sides BC, CA, AB of △ABC at D, E, F. If AF = 6, BD = 8, CE = 4, find the perimeter.

Accepted Answer

Step 1 — Equal tangents: AF = AE = 6, BD = BF = 8, CE = CD = 4.

Step 2 — AB = AF+FB = 14, BC = BD+DC = 12, CA = CE+EA = 10.

Step 3 — Perimeter = 14 + 12 + 10 = 36 cm.

✦ Answer: 36 cm.

Question 10

Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

Accepted Answer

Step 1 — Let XY be the tangent at P, centre O. Take any other point Q on XY.

Step 2 — Q lies outside the circle (only P is on it), so OQ > OP.

Step 3 — Thus OP is the shortest distance from O to XY, and the shortest distance is the perpendicular ⇒ OP ⟂ XY. ∎

✦ Answer: OP is the shortest segment, hence perpendicular.

Practice — Circles

Sign in to save attempts to your dashboard and track your accuracy over time

Find a verified tutor

Join a live course

Browse coaching centres

About

Categories

Support

Documentation

Get Study Tips & Updates

Download Our App

Download for iOS

Download for Android

Popular Cities