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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.

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. Tangents and Intersecting Chords, Concise Mathematics Solutions ICSE Class 10.

Circles

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Answer

Let AB touches the circle at L, AC at N and BC at M.

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. Tangents and Intersecting Chords, Concise Mathematics Solutions ICSE Class 10.

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 = 25\sqrt{25}

⇒ 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 = 42\dfrac{4}{2}

⇒ x = 2 cm.

Hence, x = 2.

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