A pentagon can be divided into five congruent triangles as shown. The function y = 5 tan θ models the height of each triangle. What is the area of the pentagon if θ = 54°? Round to the nearest square foot.
pentagon
(1 point)
9 ft2
86 ft2
172 ft2
344 ft2
The area of each triangle can be calculated using the formula for the area of a triangle: A = 0.5 * base * height. Since the base of each triangle is equal to the side length of the pentagon, we need to find the side length.
In a regular pentagon, the angular measure of each interior angle is 108 degrees. Since we're given that θ = 54°, each triangle covers half of that angle, so the interior angle of each triangle is 54°.
Now, we can calculate the side length of the pentagon using trigonometry. Since tan(54°) = height / length (base), we have tan(54°) = 5 tan(54°) / side length.
Solving for side length, we get:
side length = 5 tan(54°) / tan(54°) = 5
The area of the pentagon is composed of 5 congruent triangles, each with a height of 5 tan(54°) and a base (side length) of 5. Therefore, the area of the pentagon is:
5 * 0.5 * 5 * 5 tan(54°) = 25 * 5 tan(54°) = 86 ft^2
Therefore, the area of the pentagon is 86 square feet. The answer is:
86 ft^2
So, the correct answer is 86 ft^2.