Question

In the case of a triangle with angle measures of 30°, 60°, and 90° and a hypotenuse length equal to x, what is the perimeter of the triangle in terms of x?

Answer 1

Let "a" be a shorter leg (lies opposite to angle 30°), "b" is a longer leg (lies opposite to angle 60°), "c" is a hypotenuse (c = x).
In the 30°-60°-90° triangle, the side lying opposite to the angle 30° is half the hypotenuse, so:

$a= \frac{x}{2}$

By the Pythagorean theorem:

$b^2 = c^2-a^2\\ \\b^2=x^2-( \frac{x}{2} )^2 = x^2- \frac{x^2}{4} = \frac{4x^2-x^2}{4}= \frac{3x^2}{4} \\ \\ b= \sqrt{ \frac{3x^2}{4} } = \frac{ \sqrt{3x^2} }{ \sqrt{4} } = \frac{x \sqrt{3} }{2}$

$\text{Perimeter }=x+ \frac{x}{2} + \frac{x \sqrt{3} }{2}= \frac{2x+x+x \sqrt{3}}{2}= \frac{3x+x \sqrt{3} }{2}$