In the following figure Δ ABC 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.[Figu…

Answer: the tangent segment from A is 10 cm.

AB = AC = \(\dfrac{44 - 12}{2}\) = 16 cm. Let the circle touch AB at P, AC at Q and BC at R.

Tangent segments from an external point are equal: AP = AQ, BP = BR, CQ = CR.

Perimeter = (AP + PB) + (BR + RC) + (CQ + QA) = 2AP + 2BP + 2BR = 44, so AP + BP + BR = 22.

Now BP = BR and BR + RC = BC = 12 with BR = RC (by symmetry, AB = AC gives BP = BR = RC = CQ = 6 cm).

So AP = 22 – 6 – 6 = 10 cm. Hence the tangent segment from A is 10 cm.