SOLUTION: A circle has an inscribed right triangle having an altitude of 4 and an area of 28. What are the circumference and area of circle? (Inscribes means inside with vertex's touching th

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Question 37893: A circle has an inscribed right triangle having an altitude of 4 and an area of 28. What are the circumference and area of circle? (Inscribes means inside with vertex's touching the edsge of the circle)
Answer by rapaljer(4671) About Me  (Show Source):
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If a right triangle is inscribed in a circle, then the legs of that right triangle will be the base and altitude of the triangle, and the hypotenuse of the right triangle will be the diameter of the circle. To find the area and circumference of the circle, you need to find the diameter (and radius!) of the circle.
If altitude = 4 and area = 28, then
A=+%281%2F2%29%2Ab%2Ah
28+=+%281%2F2%29%2A+4%2Ah
28=2h
h=+14

Next, Let d= diameter of the circle, which is also the hypotenuse of the right triangle. According to the Theorem of Pythagoras: a%5E2+%2B+b%5E2+=+c%5E2
4%5E2+%2B+14%5E2+=+d%5E2
16+%2B+196+=+d%5E2
212+=+d%5E2
d+=+sqrt%28212%29+=+2%2Asqrt%2853%29+

Remember that the radius is HALF of the diameter, so if r= the radius, then
r=+sqrt%2853%29+ and r%5E2+=+53

Area = A=+pi%2Ar%5E2=++pi%2A53+=+53%2Api square units.

Circumference = C+=+pi%2Ad=+pi%2A+2%2Asqrt%2853%29+=2%2Api%2Asqrt%2853%29+ units.

R^2 at SCC