Question

in the diagram: - \\(\overline{eh}\\) forms the altitude for \\(\triangle ehc\\) and \\(\triangle ehi\\). - \\(\overline{ce} = 24.4\\,\text{mi}\\) - \\(\overline{hc} = 42.3\\,\text{mi}\\) - \\(\angle hie = 30.9^\circ\\) use the diagram to determine the measure of the sought sides and angles. measure of \\(\overline{eh}\\): \\(\square\\,\text{mi}\\) optional: calculate using \\(\square\\)

Explanation:

Step1: Identify the triangle type

$\triangle EHC$ is a right triangle with hypotenuse $HC = 42.3$ mi and leg $CE = 24.4$ mi. We can use the Pythagorean theorem $a^2 + b^2 = c^2$ (where $c$ is the hypotenuse, and $a,b$ are the legs) to find $EH$. Let $EH = x$, $CE = 24.4$, $HC = 42.3$. So $x^2 + 24.4^2 = 42.3^2$.

Step2: Solve for $x$

First, calculate $24.4^2 = 24.4\times24.4 = 595.36$ and $42.3^2 = 42.3\times42.3 = 1789.29$. Then, $x^2 = 42.3^2 - 24.4^2 = 1789.29 - 595.36 = 1193.93$. Now, take the square root of $1193.93$: $x = \sqrt{1193.93} \approx 34.55$ (rounded to two decimal places).

Answer:

Approximately $34.55$ mi (the value can be more precise depending on the square - root calculation method, for example, using a calculator for a more accurate $\sqrt{1193.93}$). If we use a calculator to compute $\sqrt{1193.93}$ directly, we get $\sqrt{1193.93}\approx34.55$ mi.