Question

the triangle below is equilateral. find the length of side x to the nearest tenth.

Explanation:

Step1: Recall equilateral - triangle property

In an equilateral triangle, the altitude divides the base into two equal parts. Let the side - length of the equilateral triangle be $x$. The altitude is 4, and half of the base is $\frac{x}{2}$.

Step2: Apply the Pythagorean theorem

In the right - triangle formed by the altitude, half of the base, and the side of the equilateral triangle, we have $(\frac{x}{2})^2+4^2 = x^2$.
Expand the equation: $\frac{x^{2}}{4}+16=x^{2}$.
Move the terms involving $x$ to one side: $x^{2}-\frac{x^{2}}{4}=16$.
Combine like terms: $\frac{4x^{2}-x^{2}}{4}=16$, so $\frac{3x^{2}}{4}=16$.
Cross - multiply to get $3x^{2}=64$.
Then $x^{2}=\frac{64}{3}$.
Take the square root of both sides: $x=\sqrt{\frac{64}{3}}=\frac{8}{\sqrt{3}}=\frac{8\sqrt{3}}{3}\approx4.6$.

Answer:

$4.6$