Question

the triangle below is equilateral. find the length of side x in simplest radical form with a rational denominator.

Explanation:

Step1: Identify triangle properties

The main triangle is equilateral, so all sides are 5, and the altitude (x) splits it into a 30-60-90 triangle. In a 30-60-90 triangle, the sides are in ratio \(1 : \sqrt{3} : 2\), where the side opposite 60° (x) is \(\frac{\sqrt{3}}{2}\) times the hypotenuse (5).

Step2: Calculate x

Using the ratio, \(x = 5\times\frac{\sqrt{3}}{2}=\frac{5\sqrt{3}}{2}\)? Wait, no—wait, in an equilateral triangle, the altitude (height) formula is \(h = \frac{\sqrt{3}}{2} \times \text{side length}\). Wait, the right triangle here: hypotenuse is 5 (side of equilateral), one leg is half of 5 (2.5) because the altitude bisects the base. Wait, no, the right triangle: legs are \(x\) (height) and \(\frac{5}{2}\) (half of base), hypotenuse 5. So by Pythagoras: \(x^2 + (\frac{5}{2})^2 = 5^2\).

Step3: Solve for x

\(x^2 = 25 - \frac{25}{4} = \frac{100 - 25}{4} = \frac{75}{4}\). Then \(x = \sqrt{\frac{75}{4}} = \frac{\sqrt{75}}{2} = \frac{5\sqrt{3}}{2}\). Wait, but let's check again. Wait, the equilateral triangle has side 5, so when we draw the altitude, it's a right triangle with hypotenuse 5, one leg is \(\frac{5}{2}\) (since altitude bisects the base), and the other leg is x (height). So Pythagorean theorem: \(x^2 + (\frac{5}{2})^2 = 5^2\). So \(x^2 = 25 - \frac{25}{4} = \frac{75}{4}\), so \(x = \frac{5\sqrt{3}}{2}\). Wait, but maybe I messed up the initial ratio. In 30-60-90, the sides are \(a\) (opposite 30°), \(a\sqrt{3}\) (opposite 60°), \(2a\) (hypotenuse). Here, the angle opposite x is 60°, so \(x = a\sqrt{3}\), and hypotenuse is \(2a = 5\), so \(a = \frac{5}{2}\), so \(x = \frac{5}{2}\sqrt{3} = \frac{5\sqrt{3}}{2}\). Yes, that matches.

Answer:

\(\frac{5\sqrt{3}}{2}\)