SOLUTION: A square and an equilateral triangle have equal perimeters. The area of the triangle is 9 root 3 square inches. Express, in inches, the length of a diagonal of the square.

Question 120959: A square and an equilateral triangle have equal perimeters. The area of the triangle is 9 root 3 square inches. Express, in inches, the length of a diagonal of the square.
Answer by edjones(8007) (Show Source): You can put this solution on YOUR website!
Let a=altitude, b=base A=area
A=.5ab
Construct an altitude into the equilateral triangle. Let c=hypotenuse. b=c/2.
If b=1 and c=2 then a^2+1^2=2^2
a^2=3
a=sqrt(3)
The triangle must be similar to a triangle with a base of 2 and an altitude of sqrt(3)
.
.5ab=A
.5*x*sqrt(3)*2x=9sqrt(3) The x is in there to determine how many times larger this triangle is than the 1-2-sqrt(3) similar right triangle.
x^2=9
x=3
The sides of the triangle are 2x or 6"
6*3=18 Perimeter of triangle and square.
.
side of the square=18/4=4.5"
.
a^2+b^2=c^2 (c is the diagonal of the square.)
4.5^2*2=c^2
c^2=40.5
c=6.36396"
.
Ed

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