Mathematics Question and Solution
Calculating yellow shaded area (equilateral triangle).
Let x be CE.
a = 4-3
a = 1 unit.
b²+3² = x²
b = √(x²-9) units.
b is AE.
c²+4² = x²
c = √(x²-16) units.
c is BE.
d = b+c
d = (√(x²-9)+√(x²-16)) units.
d is AB.
e²+1² = x²
e = √(x²-1) units.
e is equal d.
(√(x²-9)+√(x²-16)) = √(x²-1)
(√(x²-9)+√(x²-16))² = x²-1
x²-9+2√((x²-9)(x²-16))+x²-16 = x²-1
2√((x²-9)(x²-16)) = 24-x²
4((x²-9)(x²-16)) = (24-x²)²
4(x⁴-25x²+144) = 576-48x²+x⁴
4x⁴-100x²+576 = 576-48x²+x⁴
4x⁴-100x² = -48x²+x⁴
3x⁴ = 52x²
x² = 52/3 square units.
x² is the the square of a side length of the yellow equilateral triangle.
It implies, area yellow inscribed equilateral triangle is;
½*x²*sin60
= ½*(52/3)*½√(3)
= ⅓*13√(3)
= ⅓(13√(3)) square units.
= 7.50555349947 square units.
