Question

solve for x. leave answer in simplest radical form.
right triangle with legs 8 km and x, hypotenuse 16 km
options:
option 1: \\(\sqrt{24}\\) (or \\(2\sqrt{6}\\))
option 2: \\(\sqrt{192}\\) (or \\(8\sqrt{3}\\))

Explanation:

Step1: Identify the triangle type

The triangle is a right - triangle, so we can use the Pythagorean theorem. The Pythagorean theorem states that for a right - triangle with hypotenuse \(c\) and legs \(a\) and \(b\), \(a^{2}+b^{2}=c^{2}\). In this triangle, the hypotenuse \(c = 16\) km, one leg \(a = 8\) km, and the other leg is \(x\).

Step2: Apply the Pythagorean theorem

We know that \(a^{2}+x^{2}=c^{2}\), so we can solve for \(x\):
\[

$$\begin{align*} x^{2}&=c^{2}-a^{2}\\ x^{2}&=16^{2}-8^{2}\\ x^{2}&=256 - 64\\ x^{2}&=192 \end{align*}$$

\]

Step3: Simplify the radical

To simplify \(\sqrt{192}\), we factor 192: \(192=64\times3\). Then \(\sqrt{192}=\sqrt{64\times3}\). Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a\geq0,b\geq0\)), we have \(\sqrt{64\times3}=\sqrt{64}\times\sqrt{3}=8\sqrt{3}\).

Answer:

\(8\sqrt{3}\) (corresponding to Option 2 with the value \(8\sqrt{3}\))