Regular pentagon ABCDE is inscribed in a circle. Find measure of arch AB. If the radius of the circle is 6, find line AB.

Answer 1

you can also use the Law of Cosines to find AB

AB^2=6^2+6^2-2*6*6*cos72°

AB^2=36+36-72*0.309

AB^2=72-22.248

AB^2=49.752

AB=√(49.752)

AB=7.0535

Answer 2

Hi Maya,

The regular pentagon (5-sided polygon) divides the 360 degrees of the circle into 5 equal arcs.

That means the measure of arc AB = 360/5 = 72 degrees.

If we let O be the point at the center of the circle, then we can also draw a triangle AOB inside the circle. The central angle of this triangle (meaning the angle at point O) will be equal to the previous answer of 72 degrees.

Let M be the midpoint of AB. If we draw a line from point O to M, then we cut the triangle into two new right triangles of equal size. One of those right triangles will be AOM.

The hypotenuse of AOM is 6, since it is a radius of the circle. The central angle of AOM is 36, since it is half of 72. That means we can find the length of AM using the equation for sine:

sin = opposite / hypotenuse. Plug in the information that we know:

sin(36 degrees) = length of AM / 6

6 * sin(36 degrees) = length of AM

3.53 = length of AM

Since AM is half of AB, then line AB = 3.53 * 2 = 7.06.

Please let me know if I can be more helpful, thanks!

Joe