# 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 cmB. 2.5 cmC. 3 cmD. 3.5 cm

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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