Question

1. in a right triangle, one angle is 30°, the side adjacent to the 30° angle is $4sqrt{3}$, and the hypotenuse is $m$, the side opposite to the 30° angle is $n$. (the triangle is a right triangle with a 30° angle, right angle mark, side length $4sqrt{3}$, hypotenuse $m$, and the other leg $n$)

Explanation:

Step1: Identify the 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 side (let's say length \(x\)), the side opposite \(60^{\circ}\) is \(x\sqrt{3}\), and the hypotenuse is \(2x\).

Looking at the triangle, the side adjacent to the \(30^{\circ}\) angle (the longer leg) is \(4\sqrt{3}\), and the side opposite \(30^{\circ}\) is \(n\), the hypotenuse is \(m\).

Step2: Find the length of \(n\) (opposite \(30^{\circ}\))

Let the side opposite \(30^{\circ}\) be \(n\), the side opposite \(60^{\circ}\) (longer leg) is \(n\sqrt{3}\). We know the longer leg is \(4\sqrt{3}\), so:
\(n\sqrt{3}=4\sqrt{3}\)
Divide both sides by \(\sqrt{3}\):
\(n = \frac{4\sqrt{3}}{\sqrt{3}}=4\)

Step3: Find the length of \(m\) (hypotenuse)

In a 30 - 60 - 90 triangle, the hypotenuse \(m = 2n\). Since \(n = 4\), then \(m=2\times4 = 8\)

Answer:

If we were to find \(n\), \(n = 4\); if we were to find \(m\), \(m = 8\) (assuming we need to find the missing sides, since the problem didn't specify, but based on 30 - 60 - 90 triangle properties, these are the values).