Question

A circle is inscribed in an equilateral triangle PQR with centre 0. If Angle OQR = 30°. Find the perimeter of the triangle.

Answer 1

Answer:

Here is an equilateral triangle XYZ inscribed in a circle.

I consider the equilateral triangles to be the most balanced triangle. There sides are equal, the angles are also. The heights, medians, internal bisectors of angles are same. Therefore, the point M is a celebrity. It's incentre, circumcentre, orthocentre and centroid simultaneously. Cute Naah??

As per the diagram, MZ = 12 cm, XN is both the median and height. Centroid divides the median in 2:1 ratio. If the length of CN is x, then

2/3 × x = 12

Or, x =18

Therefore, MN = 1/3 × 18 = 6cm

As MNZ is a right triangle, so MN² + NZ²=MZ²

So, NZ = (12²-6²)^½ = 6×3^½

As N is the midpoint of YZ, YZ =2NZ=12×3^½, That's the side of this triangle.

I'm sorry for such fuzzy mathematical functions because I can't use the ‘Math’ function properly, but you'll surely get the concept.

Thanks!!

Nikuuu~