## 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

- A tangent to a circle intersects it in………points.
*Answer One* - A line intersecting a circle in two points is called a…….
*Answer Secant* - A circle can have……parallel tangents at the most.
*Answer Two* - 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

- A) 7 Cm
- B) 12 cm
- C) 15 cm
- 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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