SOLUTION: PQR is a triangle right angled at P. The lengths of the sides PQ, QR and PR are rcm p cm and q cm, respectively. Semi-circles are drawn on each side of the triangle. Show that the

Algebra ->  Pythagorean-theorem -> SOLUTION: PQR is a triangle right angled at P. The lengths of the sides PQ, QR and PR are rcm p cm and q cm, respectively. Semi-circles are drawn on each side of the triangle. Show that the       Log On


   

Question 773501: PQR is a triangle right angled at P. The lengths of the sides PQ, QR and PR are rcm p cm and q cm, respectively. Semi-circles are drawn on each side of the triangle. Show that the sum of the areas of the semi - circles, A, can be expressed as A = 1/4 π * p^2
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
I can only draw full circle. I cannot draw semicircles, so pretend you do not see the wromg half of the circles.
According to Pythagoras, r%5E2%2Bq%5E2=p%5E2.
The area of a circle is
area=pi%2Aradius%5E2
so the area of a smicircle is area=%281%2F2%29%2Api%2Aradius%5E2
The circles' diameters are r, q and p, so their radii are r%2F2, q%2F2 and p%2F2.
Then the areas of the semicircles are
%281%2F2%29%2Api%2A%28r%2F2%29%5E2, %281%2F2%29%2Api%2A%28q%2F2%29%5E2 and %281%2F2%29%2Api%2A%28p%2F2%29%5E2.
The sum of the areas of the semicircles is

Substituting p%5E2 for r%5E2%2Bq%5E2, we get
sum of semicircle areas =