Question

Chase is building a birdhouse in the shape of a regular polygon. He knows that the measure of the interior angle is twice the measure of the exterior angle and the length of a diagonal that passes through the center is

. What is the perimeter of the base of the birdhouse?

Answer 1

The perimeter of the base of the birdhouse is 36 units

Explanation:

The complete question is

Chase is building a birdhouse in the shape of a regular polygon. He knows that the measure of the interior angle is twice the measure of the exterior angle and the length of a diagonal that passes through the center is 12. What is the perimeter of the base of the birdhouse?

step 1

Find the measure of the interior angle

Let

x ---> the measure of the interior angle

y ---> the measure of the exterior angle

Remember that

the sum of the interior and exterior angle in any polygon is equal to 180 degrees

so

`x+y=180^o` ----> equation A

we have that

the measure of the interior angle is twice the measure of the exterior angle

so

`x=2y` ----> equation B

substitute equation B in equation A

`2y+y=180`

`y=60^o`

so

`x=120^o`

That means-----> The figure is a regular hexagon

step 2

Remember that

The length of the diagonal that passes through the center of the hexagon is equal to two times the length of the regular hexagon

Let

b ----> the length side of the hexagon

so

`b=12/2=6\ units`

The perimeter of the hexagon is given by the formula

`P=6b`

substitute

`P=6(6)=36\ units`