In triangle ABC, angle B = 90 degrees. Semicircles are constructed on sides AB, AC, and BC, as shown below. Show that the total area of the shaded region is equal to the area of triangle ABC.

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