Question

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?

Answer 1

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`