Question

examine the measures given in the diagram.
what is the value of y?
a. 28
b. $7\sqrt{2}$
c. $14\sqrt{3}$
d. $14\sqrt{2}$

Explanation:

Step1: Identify the triangle type

This is a right - triangle with one angle of \(30^{\circ}\), so it is a 30 - 60 - 90 right triangle. In a 30 - 60 - 90 triangle, the ratios of the sides are \(1:\sqrt{3}:2\) (opposite to \(30^{\circ}\), \(60^{\circ}\), and hypotenuse respectively). The side opposite \(30^{\circ}\) is the shorter leg, the side opposite \(60^{\circ}\) is the longer leg, and the hypotenuse is twice the shorter leg. Here, the side with length 14 is opposite the \(30^{\circ}\) angle? Wait, no. Wait, the right angle is at the bottom - right, the angle of \(30^{\circ}\) is at the bottom - left. So the side with length 14 is the opposite side to the \(30^{\circ}\) angle? No, wait, in a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, \(\theta = 30^{\circ}\), the side opposite to \(30^{\circ}\) is 14? Wait, no, wait, the side with length 14 is the vertical side (opposite to the \(30^{\circ}\) angle? Wait, no, the angle at the bottom - left is \(30^{\circ}\), the right angle is at the bottom - right. So the vertical side (height) is opposite to the \(30^{\circ}\) angle? Wait, no, the angle of \(30^{\circ}\) has the adjacent side as \(y\) (the horizontal side) and the opposite side as 14 (the vertical side). And we know that \(\tan(30^{\circ})=\frac{\text{opposite}}{\text{adjacent}}=\frac{14}{y}\)? Wait, no, that's wrong. Wait, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), so if \(\theta = 30^{\circ}\), opposite side is 14, adjacent side is \(y\)? No, wait, no. Wait, in a 30 - 60 - 90 triangle, the side opposite \(30^{\circ}\) is the shortest side. Wait, maybe I made a mistake. Let's re - consider. Let's use trigonometric ratios. We know that \(\tan(30^{\circ})=\frac{\text{opposite}}{\text{adjacent}}\), but also, \(\cot(30^{\circ})=\frac{\text{adjacent}}{\text{opposite}}\). Alternatively, we can use the fact that in a right - triangle, \(\tan(60^{\circ})=\frac{\text{opposite}}{\text{adjacent}}\). Wait, the angle at the top (the non - \(30^{\circ}\), non - right angle) is \(60^{\circ}\) because the sum of angles in a triangle is \(180^{\circ}\), so \(180 - 90 - 30=60^{\circ}\). So the side opposite \(60^{\circ}\) is \(y\), and the side opposite \(30^{\circ}\) is 14. In a 30 - 60 - 90 triangle, the side opposite \(60^{\circ}\) is \(\sqrt{3}\) times the side opposite \(30^{\circ}\). So if the side opposite \(30^{\circ}\) is 14, then the side opposite \(60^{\circ}\) (which is \(y\)) is \(14\sqrt{3}\). Wait, let's verify with trigonometric ratios. We know that \(\tan(60^{\circ})=\frac{y}{14}\), since \(\tan(60^{\circ})=\sqrt{3}\), then \(y = 14\times\tan(60^{\circ})=14\sqrt{3}\).

Step2: Calculate the value of y

Since \(\tan(60^{\circ})=\sqrt{3}\) and \(\tan(60^{\circ})=\frac{y}{14}\) (because in the right - triangle, for the \(60^{\circ}\) angle, the opposite side is \(y\) and the adjacent side is 14), we can solve for \(y\):
\[y = 14\times\tan(60^{\circ})\]
Since \(\tan(60^{\circ})=\sqrt{3}\), we have:
\[y = 14\sqrt{3}\]

Answer:

C. \(14\sqrt{3}\)