Question

question 4 (multiple choice worth 1 points) (05.01r nvc) find the value of y. image of right triangle with 60° angle, right angle, leg labeled 5√3, hypotenuse labeled y, other leg labeled x options: 5, 10, 15, 5√3

Explanation:

Step1: Identify the triangle type

This is a right - triangle with one angle \(60^{\circ}\), so the other non - right angle is \(30^{\circ}\) (since the sum of angles in a triangle is \(180^{\circ}\) and one angle is \(90^{\circ}\)). 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 call it \(a\)), the side opposite \(60^{\circ}\) is \(a\sqrt{3}\), and the hypotenuse is \(2a\).

The side with length \(5\sqrt{3}\) is opposite the \(60^{\circ}\) angle. Let the side opposite \(30^{\circ}\) be \(x\) and the hypotenuse be \(y\).

Step2: Relate the side to the ratio

We know that the side opposite \(60^{\circ}\) is \(a\sqrt{3}\), and here \(a\sqrt{3}=5\sqrt{3}\). By dividing both sides by \(\sqrt{3}\), we get \(a = 5\).

Step3: Find the hypotenuse \(y\)

The hypotenuse \(y\) in a \(30 - 60 - 90\) triangle is \(2a\). Since \(a = 5\), then \(y=2\times5 = 10\).

Answer:

10