Question

select the correct answer.
what is the value of x in the triangle?
triangle image with right angle, 30°, 60°, leg 7√3, hypotenuse? and x as the other leg
a. 21
b. 10
c. 7
d. 7√3

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 side (let's call it \(a\)), the side opposite \(60^{\circ}\) is \(a\sqrt{3}\), and the hypotenuse is \(2a\).

Step2: Determine which side is given

The side with length \(7\sqrt{3}\) is opposite the \(60^{\circ}\) angle. So, if the side opposite \(60^{\circ}\) is \(a\sqrt{3}\), and here \(a\sqrt{3}=7\sqrt{3}\), we can solve for \(a\) by dividing both sides by \(\sqrt{3}\). So, \(a = 7\).

Step3: Find the value of \(x\)

The side \(x\) is opposite the \(30^{\circ}\) angle. In a 30 - 60 - 90 triangle, the side opposite \(30^{\circ}\) is \(a\) (where \(a\) is the shortest side). Since we found that \(a = 7\), the length of \(x\) (opposite \(30^{\circ}\)) is \(7\).

Answer:

C. 7