Mathematics Question and Solution
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Calculating angle x.
Let 1 unit be the side length of the square.
Let y be the equal red lengths.
It implies;
a = (1-y) units.
b² = y²+1²
b = √(y²+1) units.
c² = 1²+1²
c² = 2
c = √(2) units.
c is the diagonal of the square.
d² = 1²+a²
d² = 1²+(1-y)²
d² = 1+1-2y+y²
d = √(y²-2y+2) units.
e = d-y
e = (√(y²-2y+2)-y) units.
Calculating y.
Observing Similar Triangles Rule.
y ~ (√(y²-2y+2)-y)
(1-y) ~ 1
Cross Multiply.
y = (√(y²-2y+2)-y)(1-y)
y = √(y²-2y+2)-y-y√(y²-2y+2)+y²
2y = √(y²-2y+2)-y√(y²-2y+2)+y²
2y-y² = √(y²-2y+2)-y√(y²-2y+2)
Squaring both sides of the derived equation.
(2y-y²)² = (√(y²-2y+2)-y√(y²-2y+2))²
4y²-4y³+y⁴ = y²-2y+2-2y(y²-2y+2)+y²(y²-2y+2)
4y²-4y³+y⁴ = y²-2y+2-2y³+4y²-4y+y⁴-2y³+2y²
0 = y²-2y+2-4y+2y²
3y²-6y+2 = 0
y²-2y = (-2/3)
(y-1)² = (-2/3)+(-1)²
(y-1)² = ⅓
y = 1±⅓√(3)
It implies;
y ≠ (1+⅓√(3)) units.
y = (1-⅓√(3)) units.
y = 0.42264973081 units.
Recall.
a = 1-y
And y = (1-⅓√(3)) units.
a = 1-(1-⅓√(3))
a = ⅓√(3) units
It implies;
tanf = a/1
tanf = ⅓√(3)/1
f = atan(⅓√(3))
f = 30°
Therefore, the required angle x is;
x = 45+f
Where f = 30°
x = 45+30
x = 75°
