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Point $Y$ lies on line segment $\overline{XZ}.$   Semicircles are constructed with diameters $\overline{XY}$, $\overline{XZ}$, and $\overline{YZ}$.  Find the area of the blue region.

 Dec 17, 2023
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Step 1: Determine XZ length

 

Using the Pythagorean theorem on triangle XYZ, we have:

XZ^2 = XY^2 + YZ^2 = 1^2 + 1^2 = 2

 

Therefore, XZ = sqrt(2).

 

Step 2: Calculate areas of semicircles

 

The radii of the semicircles are half the diameters:

 

Radius of XY semicircle = 1/2

Radius of YZ semicircle = 1/2

Radius of XZ semicircle = sqrt(2) / 2

 

Now, calculate the areas of each semicircle:

 

Area of XY semicircle = (1/2) * pi * (1/2)^2 = pi / 8

Area of YZ semicircle = (1/2) * pi * (1/2)^2 = pi / 8

Area of XZ semicircle = (1/2) * pi * (sqrt(2) / 2)^2 = (pi / 2) * (1/4) = pi / 8

 

Step 3: Find the area of the blue region

 

The blue region is the area of the XZ semicircle minus the combined areas of the XY and YZ semicircles:

 

Area of blue region = pi / 8 - (2 * pi / 8) = pi / 8 - pi / 4 = pi / 8

 

Therefore, the area of the blue region is pi/8.

 Dec 17, 2023

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