Question

a circle is inscribed in an equilateral triangle PQR with centre O. If angle OQR= 30 degree. find the perimeter of the triangle.

Answer 1

Given: A circle is inscribed in an equilateral triangle PQR with centre O and ∠OQR= 30°

To Find: perimeter of the triangle.

Solution:

Consider the following picture of a circle inscribed in an equilateral triangle:

To compare this to the question, define the notation

Q = B

R = C

P = A

Notice that angle OBC = angle OQR = 30 degrees.

Also, notice that a perpendicular line has been dropped from point O to the point D. Triangles OBD and OCD are clearly equivalent triangles (same 3 angles, and one side OD in common). Thus, D is the midpoint of BC.

Define s as the length BC. Then, s/2 is the length BD. Also, the perimeter of the triangle is clearly 3s.

Also, notice that OD = r, the radius of the circle.

Now, using the definition of tangent,

tan(30) = OD/BD = 2r/s

Thus,

s = 2r/tan(30) = 2(sqrt(3))r

and , Perimeter = 3s = 6(sqrt(3))

Perimeter = (10.39)