posted by sarah .
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.
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.
Thanks for the help.