Mathematics
ABC is a right angled triangle with AB = 12 cm and AC = 13 cm. A circle, with center O, has been inscribed inside the triangle. Calculate the value of x, the radius of the inscribed circle.
Circles
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Answer
Let AB touches the circle at L, AC at N and BC at M.
From figure,
LBMO is a square.
LB = BM = OM = OL = x.
AL = AB - LB = (12 - x) cm.
AL = AN = (12 - x) cm. [∵ Tangents from exterior point are equal in length.]
Since, ABC is a right angled triangle,
∴ AC2 = AB2 + BC2 [By pythagoras theorem]
⇒ 132 = 122 + BC2
⇒ BC2 = 132 - 122
⇒ BC2 = 169 - 144
⇒ BC2 = 25
⇒ BC =
⇒ BC = 5 cm.
From figure,
MC = BC - BM = (5 - x) cm.
Also,
CN = CM = (5 - x) cm. [∵ Tangents from exterior point are equal in length.]
Also,
⇒ AC = AN + CN
⇒ 13 = (12 - x) + (5 - x)
⇒ 13 = 17 - 2x
⇒ 2x = 17 - 13
⇒ 2x = 4
⇒ x =
⇒ x = 2 cm.
Hence, x = 2.
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