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

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2cm

2.5 cm

3 cm

3.5 cm

answer is A.

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Detailed Solution

Given figure,
Question Image

Here,

A B = 8 c m, B C = 6 cm

.
We know that according to Pythagoras theorem,

h y p o t e n u s e 2 = b a s e 2 + p e r p e n d i c u l a r 2

Consider

Δ A B C

We can observe that PBQO is a square.
Therefore, we have,
BP = BQ = QO = OP =

x .

As the lengths of tangents drawn from an external point are equal, therefore,
CR = QC, BP = BQ, and AP = AR.
Now,
AR = AP
Question Image

Similarly,
CR = QC
⇒ CR = BC - BQ
⇒ CR =

6 - x

We know that,

A C = 10 c m

⇒ A R + C R = A C

⇒ A R + C R = 10 c m ⇒ 8 − x + 6 − x c m = 10 c m ⇒ 14 − 2 x c m = 10 c m

Simplifying,

Hence, the radius of the circle i.e.,

x

2 cm

.
Therefore, option 1 is correct.

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