RS Aggarwal - Mathematics

Book: RS Aggarwal - Mathematics

Chapter: 12. Circles

Subject: Maths - Class 10th

Q. No. 8 of Exercise 12B

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8

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


Let us consider a circle with center O and XY be a tangent


To prove : Perpendicular at the point of contact of the tangent to a circle passes through the center i.e. the radius OP ⏊ XY


Proof :


Take a point Q on XY other than P and join OQ .


The point Q must lie outside the circle. (because if Q lies inside the circle, XY


will become a secant and not a tangent to the circle).


∴ OQ is longer than the radius OP of the circle. That is,


OQ > OP.


Since this happens for every point on the line XY except the point P, OP is the


shortest of all the distances of the point O to the points of XY.


So OP is perpendicular to XY.


[As Out of all the line segments, drawn from a point to points of a line not passing through the point, the smallest is the perpendicular to the line.]


Chapter Exercises

Exercise 12A
Exercise 12B
Multiple Choice Questions (MCQ)
Formative Assessment (Unit Test)

More Exercise Questions

1

In the adjoining figure, a circle touches all the four sides of a quadrilateral ABCD whose sides are AB = 6 cm, BC = 9 cm and CD = 8 cm. Find the length of side AD.
2

In the given figure, PA and PB are two tangents to the circle with center O. If ∠APB = 50° then what is the measure of ∠OAB.
3

In the given figure, O is the center of a circle. PT and PQ are tangents to the circle from an external point P. If ∠TPQ = 70°, find ∠TRQ.
4

In the given figure, common tangents AB and CD to the two circles with centers O1 and O2 intersect at E. Prove that AB = CD.
5

If PT is a tangent to a circle with center O and PQ is a chord of the circle such that ∠QPT = 70°, then find the measure of ∠POQ.
6

In the given figure, a triangle ABC is drawn to circumscribe a circle of radius 2 cm such that the segments BD and DC into which BC is divided by the point of contact D, are of lengths 4 cm and 3 cm respectively. If the area of ΔABC = 21 cm2 then find the lengths of sides AB and AC.
7

Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle (in cm) which touches the smaller circle.
8

Prove that the perpendicular at the point of contact of the tangent to a circle passes through the center.
9

In the given figure, two tangents RQ and RP are drawn from an external point R to the circle with center 0. If ∠PRQ = 120°, then prove that OR = PR + RQ.
10

In the given figure, a circle inscribed in a triangle ABC touches the sides AB, BC and CA at points D, E and F respectively. If AB = 14 cm, BC = 8 cm and CA = 12 cm. Find the lengths AD, BE and CF.
11

In the given figure, 0 is the center of the circle. PA and PB are tangents. Show that AOBP is a cyclic quadrilateral.
12

In two concentric circles, a chord of length 8 cm of the larger circle touches the smaller circle. If the radius of the larger circle is 5 cm then find the radius of the smaller circle.
13

In the given figure, PQ is a chord of a circle with center 0 and PT is a tangent. If ∠QPT = 60°, find ∠P
14

In the given figure, PA and PB are two tangents to the circle with center O. If ∠APB = 60° then find the measure of ∠OAB.

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