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△ABC is a right isosceles triangle, the centers of two arcs are the midpoint of AB
and point C. Find the difference of the area of the "moon" and the triangle, if the length of the hypotenuse is 12in.

ABC is a right isosceles triangle the centers of two arcs are the midpoint of AB and point C Find the difference of the area of the moon and the triangle if the class=

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sqdancefan

Answer:

  0

Step-by-step explanation:

Let's define 3 areas:

  • S = area of semicircle with radius 6 in (diameter AB)
  • T = area of quarter circle with radius 6√2 in (radius AC)
  • U = area of triangle ABC (side lengths 6√2)

The white space between the "moon" and the triangle has area ...

  white = T - U

Then the area of the "moon" shape is ...

  moon = S -white = S -(T -U) = S -T +U

The area we're asked to find is ...

  moon - triangle = (S -T +U) -U = S -T

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The formula for the area of a circle of radius r is ...

  A = πr²

So, ...

  S = (1/2)π(6 in)² = 18π in²

and

  T = (1/4)π(6√2 in)² = 18π in²

The difference in areas is S -T = (18π in²) -(18π in²) = 0.

There is no difference between the areas of the "moon" and the triangle.

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