# Geometry - semicircle inside isosceles triangle

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Isosceles triangle ABC has sides of length AB=AC=25 and BC=40 . Find the area of a semicircle inscribed in triangle ABC with diameter along BC .

• Geometry - semicircle inside isosceles triangle -

If the center of the semi-circle is O, draw OD where D is the point where AC is tangent to o the circle.

r^2 + a^2 =15^2
r^2 + b^2 = 20^2

equating r^2, we get
15^2-a^2 = 20^2-(25-a)^2
solve to get a=9, so b=16

So, r^2+9^2 = 15^2
r^2 = 225-81=144

area of semi-circle is 1/2 pi r^2 = 72pi

• Geometry - semicircle inside isosceles triangle -

Thanks a lot!

• Geometry - semicircle inside isosceles triangle -

Stop Cheating!

• Geometry - semicircle inside isosceles triangle -

Wow...AoPS caught you...don't know what to say...

However, it is still quite funny...

• Geometry - semicircle inside isosceles triangle -

In violation of our honor code. The previous AoPS was an imposter, his username had an uppercase "O" instead of the correct lowercase o.

• Geometry - semicircle inside isosceles triangle -

i agree- and please ask for hints on the message boards. THe questions are supposed to be challenging- try to solve them by yourself