Question

find the value of x and y.
17
x
30°
y

Explanation:

Step1: Analyze the triangle type

This is a right - triangle with one angle of \(30^{\circ}\). In a \(30 - 60-90\) right - triangle, the side opposite the \(30^{\circ}\) angle is the shortest side, and the hypotenuse is twice the length of the side opposite the \(30^{\circ}\) angle. Also, we can use trigonometric ratios (\(\sin\) and \(\tan\) or \(\cos\)) to find the sides. The side with length \(17\) is opposite the \(30^{\circ}\) angle? Wait, no. Wait, the right - angle is at the bottom - left, so the side of length \(17\) is one of the legs. Let's see: the angle of \(30^{\circ}\) is at the bottom - right. So the side opposite \(30^{\circ}\) is the vertical leg (length \(17\)), the hypotenuse is \(x\), and the horizontal leg is \(y\).

In a \(30 - 60-90\) triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\) and \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\) or \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), and also the hypotenuse \(c = 2a\) where \(a\) is the side opposite \(30^{\circ}\).

Since the side opposite \(30^{\circ}\) is \(17\), then the hypotenuse \(x = 2\times17=34\) (because in \(30 - 60 - 90\) triangle, hypotenuse is twice the side opposite \(30^{\circ}\)).

Step2: Find the length of \(y\)

We can use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = 17\), \(c=x = 34\), so \(y=\sqrt{c^{2}-a^{2}}=\sqrt{34^{2}-17^{2}}=\sqrt{(34 - 17)(34 + 17)}=\sqrt{17\times51}=\sqrt{17\times17\times3}=17\sqrt{3}\).

Or we can use the trigonometric ratio \(\tan(30^{\circ})=\frac{17}{y}\)? Wait, no. Wait, \(\tan(30^{\circ})=\frac{\text{opposite}}{\text{adjacent}}\), the opposite side to \(30^{\circ}\) is \(17\), the adjacent side is \(y\), so \(\tan(30^{\circ})=\frac{17}{y}\), then \(y=\frac{17}{\tan(30^{\circ})}\). Since \(\tan(30^{\circ})=\frac{1}{\sqrt{3}}\), so \(y = 17\sqrt{3}\). Also, using \(\cos(30^{\circ})=\frac{y}{x}\), and \(x = 34\), \(\cos(30^{\circ})=\frac{\sqrt{3}}{2}\), so \(y=x\times\cos(30^{\circ})=34\times\frac{\sqrt{3}}{2}=17\sqrt{3}\).

Answer:

\(x = 34\), \(y=17\sqrt{3}\)