# CAT Practice : Coordinate Geometry

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If one of the angles in the triangle is equal to the exterior angle of a regular hexagon and another angle is equal to the exterior angle of a regular 12-sided polygon. What can we infer?

## Area of the Triangle

Q.15: Triangle has perimeter of 6 + 2√3 . One of the angles in the triangle is equal to the exterior angle of a regular hexagon another angle is equal to the exterior angle of a regular 12-sided polygon. Find area of the triangle.
1. 2${\surd }$3
2. ${\surd }$3
3. ${\frac{√3}{2} }$
4. 3

Choice (A). 2${\surd }$3

## Detailed Solution

Given, Perimeter = 6 + 2${\surd }$3 , One of the angles in the triangle is equal to the exterior angle of a regular hexagon which is equal to 60° and another angle is equal to the exterior angle of a regular 12-sided polygon = 30°.

From this we can deduce that the other angle is equal to 90°.

The property of a 60-30-90 triangle is that, the sides are in the ratio ${\surd }$3 x, x and 2x. Therefore, Perimeter is sum of all sides = x(3+ √3)= 6 + 2√3 . => x = (6 + 2√3)/(3+ √3) = 2.

Therefore, the sides are 2${\surd }$3, 2 and 4. Area of a Right Triangle =${\frac{1}{2} }$ * Product of Perpendicular sides = ${\frac{1}{2} }$*2*2√3 = 2√3 .

Area = 2${\surd }$3.

Correct Answer: 2${\surd }$3

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