Question

1. find the missing side of the triangle. leave your answers in simplest radical form.

8 km
16 km
x

Explanation:

Step1: Identify the triangle type

This 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 \(b\), \(a^{2}+b^{2}=c^{2}\). Here, the hypotenuse \(c = 16\) km, one leg \(a = 8\) km, and the other leg is \(x\) (let's say \(b=x\)).

Step2: Substitute values into the formula

Substitute \(a = 8\), \(c = 16\) into \(a^{2}+b^{2}=c^{2}\), we get \(8^{2}+x^{2}=16^{2}\).

Step3: Simplify the equation

Calculate \(8^{2}=64\) and \(16^{2}=256\). So the equation becomes \(64 + x^{2}=256\).

Step4: Solve for \(x^{2}\)

Subtract 64 from both sides of the equation: \(x^{2}=256 - 64\). \(256-64 = 192\), so \(x^{2}=192\).

Step5: Simplify the radical

We can factor 192 as \(64\times3\). So \(x=\sqrt{192}=\sqrt{64\times3}\). Since \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a\geq0,b\geq0\)), \(\sqrt{64\times3}=\sqrt{64}\times\sqrt{3}=8\sqrt{3}\).

Answer:

\(8\sqrt{3}\) km