ΔABC is a right-angled triangle in which ∠A = 90°, AC = 12 cm and BC = 13 cm. A circle with centre O has been inscribed inside the triangle. Calculate the value of x, the radius of the inscribed circle.
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Question

ΔABC is a right-angled triangle in which ∠A = 90°, AC = 12 cm and BC = 13 cm. A circle with centre O has been inscribed inside the triangle. Calculate the value of x, the radius of the inscribed circle.

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KnowledgeBoat · Accepted Answer

Let AB touches the circle at L, AC at N and BC at M. From figure, ANLO is a square AL = LO = ON = AN = x NC = AC - AN = (12 - x) cm NC = MC = (12 - x) cm [∵ Tangents from exterior point are equal in length.] Since, ABC is a right angled triangle, ∴ BC2 = AC2 + AB2 [By pythagoras theorem] ⇒ 132 = 122 + AB2 ⇒ AB2 = 132 - 122 ⇒ AB2 = 169 - 144 ⇒ AB2 = 25 ⇒ AB = ⇒ AB = 5 cm. From figure, LB = AB - AL = (5 - x) cm. LB = BM = (5 - x) cm.[∵ Tangents from exterior point are equal in length.] Then, ⇒ BC = BM + CM ⇒ 13 = (5 - x) + (12 - x) ⇒ 13 = 17 - 2x ⇒ 2x = 17 - 13 ⇒ 2x = 4 ⇒ x = ⇒ x = 2 cm. Hence, x = 2 cm.

Mathematics

ΔABC is a right-angled triangle in which ∠A = 90°, AC = 12 cm and BC = 13 cm. A circle with centre O has been inscribed inside the triangle. Calculate the value of x, the radius of the inscribed circle.

Circles

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