Points and lie on a circle with centre

MATHEMATICS · Multi-part · CBSE · Mathematics · Circles · Circles

Problem

Points

A

and

B

lie on a circle with centre

O

. From an external point

C

, lines

CA

and

CB

are drawn so that they are tangents to the circle at

A

and

B

respectively. (i) Prove that

OA = OB

and that

OA \perp CA

and

OB \perp CB

. (ii) Let

M

be the midpoint of

AB

. Prove that the three points

C, O

and

M

are collinear. (iii) If

OA = 7\,\text{cm}

and

OC = 25\,\text{cm}

, find the length of

CM

. (Give the exact value.)