Question

In the following figure ΔABC is an isosceles triangle with perimeter 44 cm. The base BC is of length 12 cm. Side AB and AC are congruent. A circle touches the three sides of triangle as shown. Find the length of tangent segment from A to circle.

Answer 1

`{:("Let"  AP = AQ = x),(BP = BR = y),(CR = CQ = z):}}`   ...[Tangent segment theorem]

BR + RC = BC   ...(B–R–C)

∴ y + z = BC = 12   ...(1)

Similarly,

x + y = AB   ...(2)

x + z = AC   ...(3)

Adding (1), (2) and (3), we get

y + z + x + y + x + z = BC + AB + AC

∴ 2x + 2y + 2z = 44   ...[Perimeter of △ABC = 44 cm, given]

∴ 2(x + y + z) = 44

∴ `x + y + z = 44/2`

∴ x + y + z = 22   ...(4)

Substituting (1) in (4), we get,

x + 12 = 22

∴ x = 22 – 12

∴ x = 10

The length of tangent segment from A to the circle is 10 cm.