Answer:
The perimeter of the base of the birdhouse is 36 units
Step-by-step 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
[tex]x+y=180^o[/tex] ----> equation A
we have that
the measure of the interior angle is twice the measure of the exterior angle
so
[tex]x=2y[/tex] ----> equation B
substitute equation B in equation A
[tex]2y+y=180[/tex]
[tex]y=60^o[/tex]
so
[tex]x=120^o[/tex]
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
[tex]b=12/2=6\ units[/tex]
The perimeter of the hexagon is given by the formula
[tex]P=6b[/tex]
substitute
[tex]P=6(6)=36\ units[/tex]
