Question

A pentagon with 5 sides of equal length and 5 interior angles of equal measure is inscribed in a circle. Is the perimeter of the pentagon greater than 26 centimeters?(1) The area of the circle is 16π square centimeters.(2) The length of each diagonal of the pentagon is less than 8 centimeters.

Answer 1

Explanation:

Given that a pentagon with 5 sides of equal length and 5 interior angles of equal measure is inscribed in a circle

The circle has area of 16 pi sq cm.

Hence radius of circle = 4 cm

The pengagon each side would be a chord of equal length subtending angle 72 at the centre.

visualize the triangle made by one side of pentagon with centre. This is isosceles with angles 72, 54 and 54

Use sine angle for this triangle.

`(R)/(sin 54) =(s)/(sin 72)`, where R = radius =4 and s = side of pentagon.

s= 4.702

Perimeter = 5s = 23.5114 cm

Perimeter not greater than 26 cm.

If origin is at the centre then two vertices would be (4,0) (4cos 72, 4sin72),(4cos 144, 4 sin 144) and so on.

Longest diagonal = distance between (4,0) and (4cos 144, 4 sin 144)

= (-3.236, 2.351) and (4,0)

=7.608

Yes less than 8 cm