Question

o is the center of the regular pentagon below. find its perimeter. round to the nearest tenth if necessary. the area of the decagon is 60 units squared.

Explanation:

Step1: Relate decagon to pentagon triangles

A regular pentagon can be split into 5 congruent isosceles triangles from its center. A regular decagon with the same center and apothem (the 12-unit length here) is made of 10 congruent right triangles, which are exactly the halves of the pentagon's isosceles triangles. So the area of one right triangle in the decagon is $\frac{60}{10}=6$ square units.

Step2: Find base of right triangle

The apothem (12) is the height of this right triangle. Use area formula $A=\frac{1}{2}bh$:
$6=\frac{1}{2} \times b \times 12$
Solve for $b$:
$b=\frac{6 \times 2}{12}=1$

Step3: Find pentagon side length

The side length of the pentagon is twice the base of this right triangle:
$s=2 \times 1=2$

Step4: Calculate pentagon perimeter

A pentagon has 5 sides, so perimeter $P=5 \times s$:
$P=5 \times 2=10$

Answer:

10 units