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In a right triangle ABC, right-angled at B, BC = 12 cm and AB = 5 cm. The radius of the circle inscribed in the triangle (in cm) is - Mathematics

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In a right triangle ABC, right-angled at B, BC = 12 cm and AB = 5 cm. The radius of the circle inscribed in the triangle (in cm) is
(A) 4
(B) 3
(C) 2
(D) 1

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Solution

Correct answer: C

It is given that AB = 5 and BC = 12
Using Pythagoras theorem

AC2=AB2+BC2

52+122

=169

Thus AC = 13
We know that two tangents drawn to a circle from the same point that is exterior to the circle are of equal lengths.

Thus AM = AQ = a
Similarly MB = BP = b and PC = CQ = c
We know
AB = a + b = 5
BC = b + c = 12 and
AC = a + c = 13

Solving simultaneously we get a=3, b=2 and c=10
We also know that the tangent is perpendicular to the radius 
Thus OMBP is a square with side b
Hence the length of the radius of the circle inscribed in the right angled triangle is 2 cm.

Concept: Right-angled Triangles and Pythagoras Property
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