Q. 434.6( 5 Votes )

In the given figure, LABC is right - angled at B such that BC = 6 cm and AB = 8 cm. A circle with center O has been inscribed inside the triangle. OP AB, OQ BC and OR AC. If OP = OQ = OR = x cm then x = ?


A. 2 cm

B. 2.5 cm

C. 3 cm

D. 3.5 cm

Answer :

As ABC is a right triangle,


By Pythagoras Theorem


[i.e. (hypotenuse)2 = (perpendicular)2 + (base)2]


(AC)2 = (BC)2 + (AB)2


(AC)2 = (6)2 + (8)2


(AC)2 = 36 + 64 = 100


AC = 10 cm


Now,


ar(ABC) = ar(AOB) + ar(BOC) + ar(AOC)


As we know,


Area of triangle is 1/2 × Base × Height(Altitude)


Now,


OP AB [Given]


OP is the altitude in AOB


OQ BC [Given]


OQ is the altitude in BOC


OR AC [Given]


OR is the altitude in AOC


So, we have


1/2 × BC × AB = (1/2 × AB × OP) + (1/2 × BC × OQ) + (1/2 × AC × OR)


6(8) = 8(x) + 6(x) + 10(x)


[ OP = OQ = OR = x, Given]


48 = 24x


x = 2 cm

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