What is the Sum of these Angles?

Level 1

In the figure, all squares are congruent and adjacent, what is 2 x + 2 y = ? 2x+2y=? in degrees.


The answer is 90.

Hana Wehbi by Hana Wehbi , 51, USA

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2 solutions

A beautiful solution without using trigonometry can be found on Numberphile:

https://www.youtube.com/watch?v=m5evLoL0xwg

1 Helpful 0 Interesting 0 Brilliant 0 Confused

x + y = tan ⁡ − 1 ( 1 / 3 ) + tan ⁡ − 1 ( 1 / 2 ) = tan ⁡ − 1 ( 1 3 + 1 2 1 − 1 3 × 1 2 ) = tan ⁡ − 1 ( 5 6 5 6 ) = tan ⁡ − 1 ( 1 ) = 4 5 ∘ ⟹ 2 x + 2 y = 2 ( x + y ) = 9 0 ∘ x + y = \tan^{-1}(1/3) + \tan^{-1}(1/2) = \tan^{-1}\left( \dfrac{\frac{1}{3} + \frac{1}{2}}{1 - \frac{1}{3} \times \frac{1}{2}} \right) = \tan^{-1} \left( \dfrac{\frac{5}{6}}{\frac{5}{6}} \right) = \tan^{-1}(1) = 45^{\circ} \Longrightarrow 2x + 2y = 2(x + y) = \boxed{90^{\circ}} .

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Thank you, nice solution.

Hana Wehbi - 2 years, 8 months ago

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