Angle Minus Length

Geometry Level 3

Let's imagine a right triangle with its base laid flat. Let's call the base a a and the angle connecting the base and hypotenuse ∠ x ∘ \angle x^\circ . The length of the hypotenuse is equal to x x .

If the triangle has the longest base it can possibly have, what is x − a x-a ?

Note: The answer should be an integer.


The answer is 17.

Kaizen Cyrus by Kaizen Cyrus , 18, Philippines

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1 solution

Kaizen Cyrus
Aug 16, 2019

To get a a , we use the Sine Rule:

x sin ⁡ 90 ° = a sin ⁡ ( 90 − x ) ° since sin ⁡ 90 ° = 1 x sin ⁡ ( 90 − x ) ° = a \begin{aligned} \frac{x}{\sin{90°}} &= \frac{a}{\sin{(90-x)°}} & \scriptsize{\textcolor{#3D99F6}{\text{since} \space \sin{90°} = 1}} \\ & & \\ x\sin{(90-x)°} &= a & \end{aligned}

The maximum value of a a is 32.148 32.148 and for this to happen, x x must be 49.293 49.293 . Their difference would be 17 \boxed{17} if rounded to the nearest ones.

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