We have a circle with an inscribed square with an inscribed circle with an inscribed equilateral triangle. The diameter of the outer circle is 8 feet. The triangle material cost $104.95 a square foot. What is the cost of the triangular center?

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1 Answer
May 18, 2016

The cost of a triangular center is $1090.67

Explanation:

#AC = 8# as a given diameter of a circle.

Therefore, from the Pythagorean Theorem for the right isosceles triangle #Delta ABC#,
#AB = 8/sqrt(2)#

Then, since #GE = 1/2 AB#,
#GE = 4/sqrt(2)#

Obviously, triangle #Delta GHI# is equilateral.

Point #E# is a center of a circle that circumscribes #Delta GHI# and, as such is a center of intersection of medians, altitudes and angle bisectors of this triangle.
It is known that a point of intersection of medians divides these medians in the ratio 2:1 (for proof see Unizor and follow the links Geometry - Parallel Lines - Mini Theorems 2 - Teorem 8)

Therefore, #GE# is #2/3# of the entire median (and altitude, and angle bisector) of triangle #Delta GHI#.

So, we know the altitude #h# of #Delta GHI#, it is equal to #3/2# multiplied by the length of #GE#:
#h = 3/2 * 4/sqrt(2) = 6/sqrt(2)#

Knowing #h#, we can calculate the length of the side #a# of #Delta GHI# using the Pythagorean Theorem:
#(a/2)^2+h^2=a^2#
from which follows:
#4h^2=3a^2#
#a=(2h)/sqrt(3)#

Now we can calculate #a#:
#a = (2*6)/(sqrt(2)*sqrt(3)) =2sqrt(6)#

The area of a triangle is, therefore,
#S = 1/2ah = 1/2*2sqrt(6)*6/sqrt(2) = 6sqrt(3)#

At a price of $104.95 per square foot, the price of a triangle is
#P = 104.95*6sqrt(3)~~1090.67#