Bash this off?
In rectangle ABCD, AB = 8 and BC = 20. Let P be a point on AD such that ∠BP C = 90◦ .If r1, r2, r3 are the radii of the incircles of triangles AP B, BP C and CP D, what is the value of r1 + r2 + r3?
The answer is 8.
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1 solution
Hey you can do without calculation as well!! [ r 1 + r 2 + r 3 ] × 2 = [ A P + A B − B P ] + [ C P + B P − B C ] + [ C D + D P − C P ] = [ A P + A B − B C + C D + D P ] = 1 6
△ C D P is similar to △ P A B therefore 8 x = 2 0 − x 8
From which x = 4
Sides of all the triangles are then easy to get and are listed in the figure.
The formula for the radius of an inscribed circle is r = s A where A is the area of the triangle and s semi-perimeter. The individual radii then are
r 1 = 4 ( 3 − 5 ) , r 2 = 2 ( 3 5 − 5 ) , r 3 = 2 ( 3 − 5 )
r 1 + r 2 + r 3 = 8