Question

the triangle below is equilateral. find the length of side x to the nearest tenth. (there is a triangle image with a perpendicular segment of length √2 and side x marked)

Explanation:

Step1: Recall equilateral triangle height

For an equilateral triangle with side length $x$, its height $h$ is given by $h = \frac{\sqrt{3}}{2}x$.

Step2: Set height equal to given value

We know the height is $\sqrt{2}$, so set up the equation:
$$\frac{\sqrt{3}}{2}x = \sqrt{2}$$

Step3: Solve for $x$

Rearrange to isolate $x$:
$$x = \frac{2\sqrt{2}}{\sqrt{3}}$$
Rationalize the denominator:
$$x = \frac{2\sqrt{6}}{3}$$

Step4: Calculate decimal value

Compute the numerical value:
$$x \approx \frac{2 \times 2.449}{3} \approx \frac{4.898}{3} \approx 1.6$$

Answer:

$1.6$