# NCERT Solution For Class 10 Maths Circles Chapter 10

## NCERT Solution For Class 10 Maths Circles Chapter 10

In the last article for Class 10 Maths we have discussed Circles, Tangent to the circle and theorem 1.Now moving forward to this article for 10th Class Maths we will discuss theorem’s related to the Tangent.

Theorem No 2

The lengths of two tangents drawn from an external point to a circle are equal.

Proof

Given Circle with center O

Point P outside the circle

Two tangents from point P, PQ and PR

To Prove PO=PR

Let’s join OP, OQ and OR

<OQP and <ORP formed are actually right angles, as per the theorem 1, angles between the radii and tangents are right angles.

= OQ=OR                                                          (radii of same circle)

= OP=OP                                                           (Common)

= Triangle OQP=Triangle ORP                      (RHS)

= PQ=PR                                                            (Hence proved)

Let’s take an alternative method also to solve using the Pythagoras theorem to prove the same.

Alternative method (Using Pythagoras theorem)

PQ²=OP²-OQ²

=OP²-OR²=PR² (As OQ=OR)

=PQ=PR (Hence proved)

We will now solve few CBSE NCERT solutions for Class 10 Maths

Exercise 10.1

Question 1. How many tangents can a circle have?

Solution:-Infinite

Circle circumference is comprised of many infinite points, hence a circle can have infinite tangents.

Question 2. Fill in the blanks

1. A tangent to a circle intersects it in………points. Answer One
2. A line intersecting a circle in two points is called a…….Answer Secant
3. A circle can have……parallel tangents at the most. Answer Two
4. The common point of a tangent to a circle and the circle is called……Answer Point of contact

Question 3 A tangent PQ at a point P on a circle of a  radius 5 cm meets  a line through the centre  O at a point Q so that OQ=12cm,length PQ is:

A )12 cm

B )13cm

C )8.5 cm

D )√119cm

Solution:-√119cm

Exercise 10.2

Question 1 from a point Q,the length of the tangent to a circle is 24 cm and the distance of Q from the center is 25 cm.The radius of a circle is

1. A) 7 Cm
2. B) 12 cm
3. C) 15 cm
4. D) 24.5 cm

Soln:-Given PQ=24 cm, OQ=25 cm

To find OP=?

In triangle OPQ

OQ²=OP²+PQ²

OP²=OQ²-PQ²

=25²-24²

=625-576=49

OP=7cm

Question 4. Prove that the tangents drawn at the ends of a diameter of a circle are parallel?

Solution: -Construction: Let’s draw a circle with center O. Then we will draw a diameter AB. From point A and B we will draw two tangents MN and OP respectively..

To Prove:- MN||OP

<MAB=<PBA=90 degrees (radius is perpendicular to tangent)

As the alternate angles are equal

MM||OP

Hence proved.

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August 23, 2019