Question

plzz find the answer​?? plss find it

Answer 1

Question

In the given figure, line AB is tangent to both the circles touching at A and B. If OA = 29 cm, BP = 18 cm, OP = 61 cm, then find AB

$\rule{100}2$

Answer

60 cm

$\rule{100}2$

Explanation

Construction : Drop perpendicular from P on OA, let the point of intersection be Q.

Solution :

Now, angle OAB = angle PBA = 90° (Radius is always perpendicular to tangents)

And, angle AQP = 90° (by construction)

By angle sum property of a quadrilateral, angle QPB = 90° too.

Hence, AQPB is a rectangle

→ AB = PQ (Opposite sides are equal in a rectangle)

And, AQ = BP = 18 cm

Also, OQ = OA - AQ

→ OQ = 29 - 18

→ OQ = 11 cm.

Now, in ∆PQO,

PQ² + OQ² = PO² (Pythagoras theorem)

→ PQ² = PO² - OQ²

→ PQ² = 61² - 11²

→ PQ² = (61 - 11)(61 + 11)

[using, a² - b² = (a + b)(a - b)]

→ PQ² = 50 × 72

→ PQ² = 25 × 2 × 36 × 2

→ PQ = √(25 × 2 × 36 × 2)

→ PQ = 5 × 2 × 6

→ PQ = 60 cm

→ AB = 60 cm

(since, AB = PQ)