Question

find j.

write your answer in simplest radical form.
millimeters

Explanation:

Step1: Identify triangle type

This is a 30 - 60 - 90 right triangle. In a 30 - 60 - 90 triangle, the sides are in the ratio \(1:\sqrt{3}:2\), where the side opposite \(30^{\circ}\) is the shortest (let's call it \(x\)), the side opposite \(60^{\circ}\) is \(x\sqrt{3}\), and the hypotenuse is \(2x\).

Step2: Determine which side is given

The side given is \(8\sqrt{3}\) mm, which is opposite the \(60^{\circ}\) angle? Wait, no. Wait, the right angle is at the bottom right, so the side labeled \(8\sqrt{3}\) mm is adjacent to the \(30^{\circ}\) angle and opposite the \(60^{\circ}\) angle? Wait, no. Let's label the triangle: let the right angle be \(C\), \(30^{\circ}\) angle be \(A\), \(60^{\circ}\) angle be \(B\). Then side \(AC = 8\sqrt{3}\) mm (adjacent to \(30^{\circ}\), opposite \(60^{\circ}\)), side \(BC = j\) (adjacent to \(60^{\circ}\), opposite \(30^{\circ}\)), and hypotenuse \(AB\).

In a 30 - 60 - 90 triangle, \(\tan(30^{\circ})=\frac{\text{opposite}}{\text{adjacent}}=\frac{j}{8\sqrt{3}}\). We know that \(\tan(30^{\circ})=\frac{1}{\sqrt{3}}\).

Step3: Solve for \(j\)

So \(\frac{j}{8\sqrt{3}}=\frac{1}{\sqrt{3}}\). Multiply both sides by \(8\sqrt{3}\): \(j = 8\sqrt{3}\times\frac{1}{\sqrt{3}}\). The \(\sqrt{3}\) terms cancel out, so \(j = 8\).

Wait, another way: In 30 - 60 - 90 triangle, the side opposite \(30^{\circ}\) is \(x\), opposite \(60^{\circ}\) is \(x\sqrt{3}\). Here, the side opposite \(60^{\circ}\) is \(8\sqrt{3}\), so \(x\sqrt{3}=8\sqrt{3}\), which means \(x = 8\). And the side opposite \(30^{\circ}\) (which is \(j\)) is \(x\), so \(j = 8\).

Answer:

\(8\)