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# Semicircle

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A semicircle is inscribed in triangle $XYZ$ so that its diameter lies on $\overline{YZ}$, and is tangent to the other two sides.   If $XY = 10,$ $XZ = 10,$ and $YZ = 10 \sqrt{2},$ then find the area of the semicircle.

Jan 13, 2024

#1
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My idea: Reflect triangle XYZ across line segment YZ, to form a square, which is possible because this is a 45 45 90 triangle where YZ is the hypotenuse. We can find the area of the semicircle by finding the area of the circle and then dividing it by 2.

The diameter of the circle will be 10 because it is inscriibed in the square of length 10, so the radius is 5.

using A = pir^2, we get 25 pi, or 12.5 pi for the semicircle.

We can approximate 12.5pi as 36.27

Jan 13, 2024
#2
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Let O be the center of the semi-circle

Let P be the tangent  point of  the semi-circle with the  triangle

Connect OP  = the radius of the semi-circle

We have triangle  OPZ

The triangle is equilateral so  angle PZO = 60

Angle OPZ = 90   (a radius meeting a tangent  forms a  90 degree angle )

So angle POZ = 30

We have a 30-60 -90  right triangle

OZ   = (1/2)YZ = (1/2) (10)  =  5

PZ = 5/2

So  OP =  (5 /2)sqrt (3)     = the radius of the  semi-circle

Area of semi-circle   = (1/2 pi * [ ( 5/2)sqrt 3 )  ] ^2 =   (75 / 8)  pi

Jan 13, 2024