Question

An isosceles triangle has two 10.0-inch sides and a 2w-inch side. Find the radius of the inscribed circle of this triangle, in the cases w = 5.00, w = 6.00, and w = 8.00.
Then Write an expression for the inscribed radius r in terms of the variable w , then find the value of w, to the nearest hundredth, that gives the maximum value of r.

Answer 1

The inscribed circle has its centre on the bisectors of the angles.
Because of the properties of isosceles triangles that angle bisector also becomes the right-bisector of the non-equal side, or our 2w base.

I will do the w=8 case.

draw a 10-10-16 triangle, 16 as the base
label the base angle 2Ø, thus each of the bisected base angles are Ø.
label the length of the radius on the right-bisector as r, (where the angle bisector meets the righ-bisector of 16)
cos 2Ø = 8/10 = 4/5
we know cos 2Ø = 2cos^ Ø - 1
4/5 = 2cos^2 Ø -
cos^2Ø = (4/5 + 1)/2 = 9/10
cosØ = 3/√10
then sinØ = 1/√10 , using Pythagoras

back to the triangle, in the smaller right-angled triangle
tanØ = r/8
sinØ/cosØ = r/8
(1/√10)/(3/√10) = r/8
1/3 = r/8
r = 8/3

Now you repeat the calculations for w = 6 and see if you can see the pattern.
then generalize, or go through the same process, for w.

Answer 2

Thanks for the help.