Exercise 10.1

1. How many tangents can a circle have?

Solution:

A tangent to a circle is a line that intersects the circle at only oneonly two point.

On every point on the circle, one tangent can be drawn as shown in the figure below.

As per the above diagram, we see that a circle can have infinitelyfinite many tangents.

As the circle is a set of points equidistant from a fixed point, there will be infinitefinite points that form a circle and hence, infinitefinite tangents are possible.

2. Fill in the blanks :

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

(ii) A line intersecting a circle in two points is called a .

(iii) A circle can have parallel tangents at the most.

(iv) The common point of a tangent to a circle and the circle is called point of contactpoint of intersection.

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. Find the Length of PQ?

Solution:

Given: Radius OP = cm, OQ = cm

We have to find the length of the tangent PQOQ.

ΔOPQ is a right-angle triangle according to the Theorem 10.1.

By Pythagoras theorem:

OQ2 = OP2 + PQ2QP2

122132 = 5272 + PQ2

= + PQ2

PQ2 =

PQ = 119119

Thus, the length of PQ is 119 cm.

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.

Start by joining the center of the circle to a point M, in the interior of the circle.

Now, join the center with a point P, lying on the circumference of the circle.

Solution:

Tangent line or tangent in geometry means a line or plane that intersects a curve or a curved surface at exactly point.

Secant in geometry means a line that intersects points on an arc/circumference of a circle.

As shown in the diagram above, XY is the given line.

AB is the parallel to XY, AB || XY.

PQ is the parallel to XY, PQ || XY.