# CAT Practice : Coordinate Geometry

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What do you need to find, to calculate the area of a triangle? Determine its base and altitude!

## Circles touching externally

Q.30: Two circles with centres O1 and O2 touch each other externally at a point R. AB is a tangent to both the circles passing through R. P’Q’ is another tangent to the circles touching them at P and Q respectively and also cutting AB at S. PQ measures 6 cm and the point S is at distance of 5 cms and 4 cms from the centres of the circles. What is the area of the triangle SO1O2?
1. 9 cm2
2. 3(4 + ${\surd 7}$)/2 cm2
3. 27/2 cm2
4. (3${\surd 41}$)/2 cm2

Choice (B). 3(4 + ${\surd 7}$)/2 cm2

## Detailed Solution

From the diagram we see that SP, SR are tangents to circle1 from same point S. Similarly SR, SQ are tangents from same point to circle 2.

SP = SR; SQ = SR implies SP = SQ

Given PQ = 6cm

SP + SQ = 6

Therefor SR = SP = SQ = 3 cm.

SR is the altitude to the triangle SO1O2. We need to find the length of the base O1O22 to determine the area.

O1RS is a right angled triangle with hypotenuse = 5 and one side = 3

Therefore, O1R = ${\surd (5^2 - 3^2)}$ = 4 cm

Similarly, O2RS is a right angled triangle with hypotenuse = 4 and one side = 3

Therefore, O2R = ${\surd (4^2 - 3^2)}$ = ${\surd 7}$ cm

O1O2 = O1R + O2R = 4 + ${\surd 7}$

Area of the triangle SO1O2 = 1/2 * SR * O1O2 = 1/2 * 3 * (4 + ${\surd 7}$) cm2.

Correct Answer: (B). 3(4 + ${\surd 7}$)/2 cm2

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## More questions from Geometry Triangles

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.