What is the radius of the circle inscribed in triangle ABC if AB = 5, AC=7, BC=8? Express your answer in simplest radical form.
+1641 What is the radius of the circle inscribed in triangle ABC if AB = 5, AC=7, BC=8? Express your answer in the simplest radical form.
r = √3 area of ΔABC = 10r area of a circle is 3pi 
+129907 Let B =(0,0) C = (8,0)
And we can find the x coordinate of A by letting the circles x^2 + y^2 = 25 and (x - 8)^2 + y^2 = 49 intersect
Subbing y^2 = 25 -x^2 into the second equation we have that
(x - 8)^2 + 25 - x^2 = 49
x^2 - 16x + 64 + 25 - x^2 = 49
-16x + 89 = 49
16x = 40
x = 40/16 = 2.5 ⇒ x^2 = 6.25
And y = sqrt (25 - 6.25) = sqrt [ 18.75 ] = sqrt (75/4) = sqrt (25 * 3) / 2 = (5/2) sqrt (3)= the altitude of this triangle
The area of this triangle = (1/2)( BC) (altitude) = (1/2) (8) (5/2) sqrt (3) = 10sqrt (3)
We can find the radius,R, of the inscribed circle thusly
(1/2) R ( 8 + 7 + 5 ) = 10sqrt (3)
10R = 10sqrt (3)
R = sqrt (3)
