Question

Triangle has peremeter of 6+2√3. one of
angle in triangle is equal to exterior
angle of regular hexagon another angle
is equal to exterior angle of regular
12-sided polygon. find the area of triangle?​

Answer 1

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 "Find the area of the triangle"

Hence, the answer is 2√3