The vertices of an equilateral triangle, with perimeter P and area A, lie on a circle with radius r. Find an expression for P/A in the form K/r, where k ∈ Z+
Draw the circle. Draw its center. Label the center as "M".
Draw the triangle inside. Note that all three angles have the same measure, and all three sides have the same length (being P/3). Draw the height line from one vertex down to the midpoint of the opposite side, which is the base of the triangle. Draw a radius line from the center (which is on the height line) to one of the base angles of the triangle.
Label the peak angle (at the "top" of the triangle) as "A", the vertex at the left as "B", the vertex at the right as "C", the point where the height line meets the base as "D", the height line as "h", the side AB as "P/3", and the radius line as "r". (This way, we'll all be looking at the same picture.)
What is the length of the base of the right triangle ADB? Label this. You already have the length of AB. Now use the Pythagorean Theorem to find the length of h. You know that the length of AM is r. What then is the length of MD? Does this information help you toward your goal?
You have a smaller right triangle, MBD. You know that the hypotenuse MB has length r. You know the length of BD in terms of P. Can you use the Pythagorean Theorem to solve the triangle? Can you find two expressions you can set equal, or something that you can solve for P in terms of r?
Can you find a way of expressing the area of the triangle, or some portion of the triangle, in terms of r?
Please reply showing your work so far. Thank you!
