Hi! I'm currently stuck on solving this problem:

"The triangle ABC is an isosceles triangle where AB = 4sqrt(2) and angle B is a right angle. If I is the incenter of triangle ABC, then what is BI? Express your answer in the form a + b*sqrt(c) where a, b, and c are integers, and c is not divisible by any perfect square other than 1."

For this problem, I drew triangle ABC with angle B as the right angle, the incircle, and the angle bisectors, and dropped down lines from the incenter to the points of tangency the circle had with the triangle. I knew those three dropped-down lines all had a length of the inradius and that they were perpendicular to a side of the triangle ABC. Thus, I realized that BI was a diagonal of a square with side lengths of the inradius, which meant that BI was equal to sqrt(2r^2) = r*sqrt(2), where r is the inradius. However, I didn't know what to do next or how to find r. Can someone give me a hint? I don't want the answer; I just want to know how to solve the problem. Thank you so much!

Guest Nov 21, 2020

edited by
Guest
Nov 21, 2020

edited by Guest Nov 21, 2020

edited by Guest Nov 21, 2020

#1**+2 **

We have an isosceles right triangle..... legs = 4sqrt (2) hypotenuse = 8

Perimeter = 8 + 8sqrt (2) = 8 ( 1 + sqrt (2) )

Let B = (0,0)

Let A = (0, 4sqrt(2) )

C = (4sqrt(2) ,0)

Formula to find the incenter =

x = ( side opposite A * x coordinate of A + side opp B * x coordinate of B + side opp C * x coordinate C) / perimeter

So we have

x = ( 4sqrt (2) * 0 + 8 * 0 + 4sqrt (2)*4sqrt (2) )/ [ 8 (1 + sqrt (2))] =

(32) / [ 8 (1 + sqrt (2)] =

4/ ( 1 + sqrt (2) ) = ( multiply by 1 -sqrt (2) on top/bottom and simplify ]

4 ( 1 - sqrt (2) ) / (-1) =

4 (sqrt (2) - 1 ) = 4sqrt (2) - 4 = sqrt (32) - 4

y = ( side opposite A *y coordinate of A + side opp B * y coordinate of B + side opp C * y coordinate of C) /perimeter

If you do this correctly you will find that x = y = sqrt (32) - 4

BI = sqrt ( [ sqrt (32) - 4 - 0 ]^2 + [ sqrt (32) - 4 - 0]^2 ) =

sqrt [ 32 - 8sqrt (32) + 16 + 32 - 8sqrt (2) + 16 ] =

sqrt [ 96 - 16 sqrt (32) ] =

sqrt [ 16 (6 - sqrt (32) ] =

4 sqrt [ 6 - 4sqrt (2 ] ≈ 2.343 units

See the following image :

i is the intersection of the angle bisectors......

CPhill Nov 21, 2020