Exercise 53 Page 722

We are given that ∘ P is inscribed in equilateral triangle LMN and we want to find the circumference of the circle. To do this we need to find the radius of the circle. Let's call it r.

Now notice that we can divide an equilateral triangle into six congruent 30^(∘)-60^(∘)-90^(∘) triangles. Let's connect the vertices with the center of the circle and draw radii to the points of tangency.

In a 30^(∘)-60^(∘)-90^(∘) triangle the length of the longer leg is sqrt(3) times the length of the shorter leg. rsqrt(3)=4 ⇒ r=4sqrt(3)/3 The radius of the circle is 4sqrt(3)3. Now let's recall that the circumference of a circle is two times the product of the radius and pi. C=2π* 4sqrt(3)/3=8π sqrt(3)/3 The circumference of ∘ P is 8πsqrt(3)3.