Question

In the figure given below, find RS and PS using the information given in \(\triangle\)PSR. 

Hint:

The 30-60-90 triangle theorem is a very powerful and quick tool for solving such problems. Recognizing this special triangle saves you from having to perform trigonometric calculations.

Answer 1

Step 1: Understanding the Concept:
This problem involves solving a right-angled triangle using trigonometric ratios or the properties of a 30-60-90 triangle. Given one side and one angle, we can find the lengths of the other two sides.

Step 2: Key Formula or Approach:
We can use two methods:
Method 1: Trigonometric Ratios \[\begin{array}{rl} \bullet & \text{\( \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \)} \\ \bullet & \text{\( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)} \\ \end{array}\] Method 2: 30-60-90 Triangle Theorem The sides of a 30-60-90 triangle are in the ratio \( 1 : \sqrt{3} : 2 \). \[\begin{array}{rl} \bullet & \text{The side opposite 30\(^\circ\) is half the hypotenuse.} \\ \bullet & \text{The side opposite 60\(^\circ\) is \( \sqrt{3} \) times the side opposite 30\(^\circ\).} \\ \end{array}\]

Step 3: Detailed Explanation:
Given: \[\begin{array}{rl} \bullet & \text{In \(\triangle\)PSR, \(\angle\)S = 90\(^\circ\).} \\ \bullet & \text{Hypotenuse PR = 12.} \\ \bullet & \text{\(\angle\)P = 30\(^\circ\).} \\ \bullet & \text{Therefore, \(\angle\)R = 180\(^\circ\) - 90\(^\circ\) - 30\(^\circ\) = 60\(^\circ\).} \\ \end{array}\] Using Method 2 (30-60-90 Triangle Theorem):
1. Find RS: RS is the side opposite the 30\(^\circ\) angle (\(\angle\)P). According to the theorem, the side opposite the 30\(^\circ\) angle is half the hypotenuse. \[ RS = \frac{1}{2} \times PR = \frac{1}{2} \times 12 = 6 \] 2. Find PS: PS is the side opposite the 60\(^\circ\) angle (\(\angle\)R). According to the theorem, the side opposite the 60\(^\circ\) angle is \( \sqrt{3} \) times the side opposite the 30\(^\circ\) angle. \[ PS = \sqrt{3} \times RS = \sqrt{3} \times 6 = 6\sqrt{3} \]

Step 4: Final Answer:
The length of RS is 6 and the length of PS is 6\(\sqrt{3}\).