# CAT Practice : Coordinate Geometry

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Radius of the circle inscribed inside a semi - circle

## Circles

Q.38: Two circles of radius 5 cm have a direct tangent PQ and an indirect tangent RS. Find the length of PQ if RS = 24 cm.
1. 29 cm
2. 13 cm
3. 26 cm
4. Cannot be determined due to lack of information

Choice (C). 26 cm

## Detailed Solution

Let us draw the required diagram as:-

Since, OP = PQ and OP⊥PQ and O’Q⊥PQ
PQ = OO’
Now in △ORT and △O’ST
∠ORT = ∠O’ST
∠OTR = ∠STO’
∴ △ORT is similar to O’ST (AA)
${\frac{OR}{OS}} = \frac{RT}{TS} =) \frac{RT}{TS} = \frac{5}{5} = 1$
=) RT = TS
Given RS = 24 cm
∴ RT = TS = 12cm
∴ OT = √((OR)2 + (RT)2 ) = √(52 + 22 ) = 13 cm = O’T
∴ OO’ = PQ = 13 +13 = 26 cm = (c) Ans.

Correct Answer: (C). = 26 cm

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Geometry is probably the most vital topic as far as CAT preparation is concerned. Geometry sets the stage for Trigonometry, Cogeo and Mensuration as well.