lie on the circumference of a circle of radius
* A regular pentagon is such that its vertices
4.5 cm. Find the length of a side of the
pentagon to the nearest mm.
Answers
Answer:
A regular pentagon means that sides and angles are equal. The measure of an external angle is 360/n. n=5 so the measure of one exterior angle is 72 degrees. Then, the measure of one of the interior angles is 180 - 72 = 108 degrees.
a) The sides of the triangles ending at the vertices of the pentagon will bisect the angles. Also, these triangles will be isosceles because the equal sides will be the radius of the circle. So, we will have two base angles, each measuring 108/2 = 54 degrees. Then, the central angle will measure 180 - (2*54) = 72 degrees
b) If you draw a perpendicular line from the center to one of the bases of these triangles formed, Moreover, it will bisect the central angle and also the base, which is the side of the pentagon. We will have two congruent right triangles with angles 54 and 36 degrees. Let the length of the pentagon's side be 2x. Then, the length of the side of the right triangles opposite 36 degrees will each be x. Using trigonometry, sin(36 deg) = x/5 and we get x = 2.9 (rounded to the nearest tenth) Multiply by 2 to get the pentagon's side length. 2*2.9 = 5.8 inches.
c) The perimeter of the pentagon is 5*5.8 = 29 inches
d) The area of the pentagon will be the sum of the five congruent triangles formed. The area of one of these triangles is equal to the side of the pentagon*height/2. We need to find the height. We can easily find it from the relationship we found in part b. Using one of those right triangles formed, we can write cos(36 deg) = h/5 and we get h = 4.0 inches (rounded to the nearest tenth).
area of one triangle = (5.8*4.0)/2 = 11.6 in^2 (rounded to the nearest tenth)
Multiply by 5 to get the area of the pentagon. 11.6*5 = 58 in^2
Given : a regular Pentagon whose vertices lie on the circumference of a circle of radius 5 cm.
To Find : length of a side of Pentagon
Solution:
Central angle = 360°/5 = 72°
Radius = 5 cm
using cosine rule
a² + b² - 2abCosC = c²
Side of pentagon = x
=> x² = r² + r² - 2 r*rCos72°
=> x² =2r² - 2r²Cos72°
=> x² =2r²(1 - Cos72°)
=> x =1.17557 * r
=> x = 1.17557 * 5
=> x = 5.87785
length of a side of Pentagon = 5.87785 cm
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