A bit of Trigonometry

Geometry Level 2

In the image, BDG is a rectangular triangle with side lengths of 3, 4 and 5. ABCD, DGEF and BIGH are squares.

Find x² + y².


The answer is 125.

Franklin Pedro Da Silva Neto by Franklin Pedro Da Silva … , 25

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2 solutions

Aly Ahmed
Jan 11, 2017

1 Helpful 0 Interesting 0 Brilliant 0 Confused

is there any other method than this.

Sathvik Acharya - 4 years, 3 months ago

Consider the diagram on the left.

sin ⁡ θ = 3 5 \sin \theta=\dfrac{3}{5} ⟹ \implies θ = sin ⁡ − 1 ( 3 5 ) ≈ 36.87 \theta = \sin^{-1}\left(\dfrac{3}{5}\right) \approx 36.87

β = 90 − 36.87 ≈ = 53.13 \beta = 90 - 36.87 \approx = 53.13

α = 360 − 90 − 36.87 − 90 ≈ 143.13 \alpha=360-90-36.87-90 \approx 143.13

ϕ = 360 − 90 − 53.13 − 90 ≈ 126.87 \phi = 360-90-53.13-90 \approx 126.87

By law of cosines, x 2 = 4 2 + 5 2 − 2 ( 4 ) ( 5 ) ( cos ⁡ 143.13 ) = 73 x^2=4^2+5^2-2(4)(5)(\cos 143.13)=73 ; y 2 = 3 2 + 5 2 − 2 ( 3 ) ( 5 ) ( cos ⁡ 126.87 ) = 52 y^2=3^2+5^2-2(3)(5)(\cos 126.87)=52 .

x 2 + y 2 = 73 + 52 = 125 x^2+y^2=73+52=125

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