Question

An equilateral triangle of side length 30 is inscribed in a ˚le. What is the radius of the ˚le? Your answer should be in the form a√b where a and b are both integers.

(a) 15√3
(b) 10√3
(c) 5√3
(d) 20√3

Answer 1

To calculate the radius of the circle, an equilateral triangle's side length was divided by the square root of 3, then multiplied by the height of a 30-60-90 right triangle, resulting in a radius of c) 5√3.

Step-by-step explanation:

To find the radius of the circle in which an equilateral triangle of side length 30 is inscribed, we can make use of the Pythagorean theorem. Given that each angle in an equilateral triangle is 60 degrees, we can draw a radius from the center of the circle to one of the vertices of the triangle, creating two 30-60-90 right triangles.

In a 30-60-90 triangle, the lengths of the sides are in the ratio of 1:√3:2. Therefore, if the side opposite the 60-degree angle (which is half of the side of the equilateral triangle) is 15, then the side opposite the 30-degree angle (which is the radius we are trying to find) will be 15/√3. To make the radius the subject, we multiply both sides by √3 to get the radius by itself, giving us a radius, r, of 15*√3/3 = c)5√3.