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Chapter: Circle· Mathematics Part 2 (Geometry)

Asked in: July 2025 board paperIn the following figure \(\Delta ABC\) 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…

Asked in: July 2025 board paper

In the following figure \(\Delta ABC\) 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.

[Figure: the incircle of \(\Delta ABC\) touches AB at P, BC at Q and AC at R.]

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Answer: 10 cm

AB + AC + BC = 44 and BC = 12 \(\therefore AB + AC = 32\). Since AB = AC, AB = AC = 16 cm.

Let the circle touch AB at P, BC at Q and AC at R. Tangent segments from an external point are congruent: AP = AR, BP = BQ, CQ = CR.

Let AP = AR = x. Then BP = 16 − x and CR = 16 − x, so BQ = CQ = 16 − x.

BQ + CQ = BC \(\therefore 2(16 - x) = 12\) \(\therefore 16 - x = 6\) \(\therefore x = 10\).

So the tangent segment from A to the circle is 10 cm.

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