SOLUTION: HELPPPP... Square WXYZ is inscribed in the circle O. If the area of the triangle OZY = 8 what is the area of shaded region?

Question 896471: HELPPPP...

Square WXYZ is inscribed in the circle O. If the area of the triangle OZY = 8 what is the area of shaded region?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
from the diagram and the information supplied below, you should be able to figure out what you need.

from the diagram.

area of each of the triangles = A
area of each of the segments of the circle outside the square = B
area of each of the sectors of the circle = A + B
area of the circle = 4 * (A + B)
area of the square = 4 * A
side of the square = S
radius of the circle = R

values for information supplied in the diagram are:

A = 8
B = 4*pi - 8
A + B = 4*pi
4 * (A + B) = 16*pi
4 * A = 32
S = sqrt(32)
R = 4
pi = 3.141592654

you should be able to figure out what you need from the diagram and the supplied information.

the diagram is shown below:

$$$

the length of a side of the square was derived as follows:

each of the triangle is congruent, therefore the area of the square = 4 * 8 = 32
the area of the square = S^2 therefore S = sqrt(32)

the diagonal of the square is equal to sqrt(sqrt(32)^2 + sqrt(32)^2) which is equal to sqrt(32+32) which is equal to sqrt(64) which is equal to 8.

half the diagonal of the square is equal to the radius so the length of the radius is equal to 4.

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