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The question is from CAT Geometry - Triangles. In this problem perimeter of a triangle and hints about angles are given and we have to find out the area of the triangle. CAT Geometry questions are heavily tested in CAT exam. Make sure you master Geometry problems. 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?

Question 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√3
  2. √3
  3. √3/2
  4. 3

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Explanatory Answer

Method of solving this CAT Question from Triangles: Finding out the angle using the hints is the key idea here.

Given, Perimeter = 6 + 2√3
One of the angles in the triangle is equal to the exterior angle of a regular hexagon which is equal to 60°
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 √3x, 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√3, 2 and 4.
Area of a Right Triangle = 1/2 * Product of Perpendicular sides = 1/2 * 2 * 2√3 = 2√3 .

The question is "Perimeter of a △ with integer sides is equal to 15. How many such triangles are possible?"

Hence, the answer is 2√3.

Choice A is the correct answer.

 

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