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**CLASS 10 MATHEMATICS**

### Ch 10 – CIRCLES

Ex 10.1

**1. How many tangents can a circle have?**

**Answer:**

A circle can have **infinite** tangents to a circle.

We know that A circle is made up of infinite points which are at an equal distance from a point.

So there are infinite points on the Circumference of a circle,

Infinite tangents can be drawn from them.

**2. Fill in the blanks:**

**(i) A tangent to a circle intersects it in**

** …………… point(s).**

Answer (i)

A tangent to a circle intersects it in **one** point(s).

**(ii) A line intersecting a circle in two points**

** is called a ………….**

Answer (ii)

A line intersecting a circle in two points is

called a **secant.**

**(iii) A circle can have …………… parallel**

** tangents at the most.**

**Answer: **(iii)

A circle can have **two **parallel tangents at the most.

**(iv) The common point of a tangent to a circle**

** and the circle is called …………**

**Answer** (iv)

The common point of a tangent to a circle and the circle is called the **point of contact.**

**3. A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at** **a point Q so that OQ = 12 cm. Length PQ is :**

**(A) 12 cm**

**(B) 13 cm**

**(C) 8.5 cm**

**(D) √119 cm**

**Answer:**

In this figure,

the line OP is perpendicular to tangent PQ.

i.e. OP ⊥ PQ

Using Pythagoras theorem in triangle ΔOPQ,

Where Angle P = 90

i.e. (Hypotenuse)^{2}= (base)^{2}+ (perpendicular)^{2}

thus,

OQ^{2} = OP^{2}+PQ^{2}

(12)^{2 }= 5^{2}+PQ^{2}

144 = 25 + PQ^{2}

PQ^{2} = 144-25

PQ^{2} = 119

PQ = √119 cm

Therefore,

**option D** i.e. √119 cm is the length of PQ.

**4. Draw a circle and two lines parallel to**

** a given line such that one is a tangent and the**

**other, a secant to the circle.**

**Answer: **

In the figure,

XY and AB are the two parallel lines.

The line segment AB is the tangent at point C

while the line segment XY is the secant.