Mathematics Question and Solution
Calculating the area of the equilateral triangle (one of the two equal green inscribed equilateral triangles).
Let x be the side length of the two equal green inscribed equilateral triangles.
a = ½(8)
a = 4 units.
b² = ½(10)
b = 5 units.
c = ½(10-8)
c = ½(2)
c = 1 units.
d²+4² = x²
d = √(x²-16) units.
e²+5² = x²
e = √(x²-25) units.
f = d+e
f = (√(x²-16)+√(x²-25)) units.
g²+1² = x²
g = √(x²-1) units.
Notice.
f = g
Calculating x.
(√(x²-16)+√(x²-25)) = √(x²-1)
(√(x²-16)+√(x²-25))² = x²-1
x²-16+2(√((√(x²-16)√(x²-25)))+x²-25 = x²-1
2(√((x²-16)(x²-25)) = 40-x²
2²((x²-16)(x²-25)) = (40-x²)²
4(x⁴-41x²+400) = 1600-80x²+x⁴
4x⁴-164x²+1600 = 1600-80x²+x⁴
3x⁴ = 84x²
x² = 84/3
x = √(28)
x = 2√(7) units.
x = 5.29150262213 units.
Again, x is the side length of the two equal green inscribed equilateral triangles.
It implies, area green inscribed equilateral triangle is;
½*x²sin60
= ½*2√(7)*2√(7)*½√(3)
= 7√(3) square units.
