Explanation:
The area of a semicircle is given by ...
A = πr^2/2
where r is the radius. Here, we're given diameters, so in terms of diameter, the area of a semicircle is ...
A = π(d/2)^2/2 = (π/8)d^2
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The area of the semicircle with diameter AC is ...
white area = (π/8)AC^2
The area of the semicircle with diameter BC is ...
left semicircle area = (π/8)BC^2
And the area of the semicircle with diameter AB is ...
right semicircle area = (π/8)AB^2
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We can use the relationship between the areas to find the shaded area:
triangle area + left semicircle area + right semicircle area =
white area + shaded area
Then the shaded area is ...
shaded area = triangle area + left semicircle area + ...
right semicircle area - white area
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Filling in the values for area from above, we have ...
shaded area = triangle area+ (π/8)BC^2 +(π/8)AB^2 -(π/8)AC^2
shaded area = triangle area + (π/8)(BC^2 +AB^2 -AC^2)
From the Pythagorean theorem, we know that ...
AC^2 = BC^2 +AB^2
Substituting this into the above equation gives ...
shaded area = triangle area + (π/8)((Bc^2 +AB^2 -(BC^2 +AB^2))
shaded area = triangle area + 0 . . . . simplify
shaded area = triangle area
