Question

question 26 of 50
the value of x is:
right triangle with right angle, one leg 11, one acute angle 30°, the other leg x
options: 6.4, 11, 22

Explanation:

Step1: Identify triangle type

This is a right - triangle with one angle \(30^{\circ}\), one angle \(90^{\circ}\), so the third angle is \(60^{\circ}\). In a \(30 - 60 - 90\) right - triangle, we know the trigonometric relationship \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 30^{\circ}\), the adjacent side to \(30^{\circ}\) is \(11\), and the opposite side is \(x\). So \(\tan(30^{\circ})=\frac{x}{11}\).
We know that \(\tan(30^{\circ})=\frac{1}{\sqrt{3}}\approx0.577\). Then \(x = 11\times\tan(30^{\circ})\approx11\times0.577\approx6.35\approx6.4\).
Alternatively, we can also use the ratio of sides in a \(30 - 60 - 90\) triangle. The sides are in the ratio \(1:\sqrt{3}:2\) (opposite to \(30^{\circ}\), \(60^{\circ}\), \(90^{\circ}\) respectively). But in this case, the adjacent side to \(30^{\circ}\) is \(11\), and we are finding the opposite side. Using \(\tan\) is more direct here.

Step2: Calculate the value of x

Using the formula \(x = 11\times\tan(30^{\circ})\), and \(\tan(30^{\circ})=\frac{\sqrt{3}}{3}\approx0.577\).
\(x = 11\times\frac{\sqrt{3}}{3}=\frac{11\sqrt{3}}{3}\approx\frac{11\times1.732}{3}=\frac{19.052}{3}\approx6.35\approx6.4\)

Answer:

6.4