Question

If the side opposite to ${{30}^{\circ }}$angle in a right triangle is 12.5 meters, how long is the hypotenuse?

Answer 1

Hint: To solve this question, we should know the trigonometric ratios, and how to find their values for a given triangle. We have to find the length of the hypotenuse. We will use the sine ratio in trigonometry for this. In a right angled triangle, hypotenuse is the longest side and is opposite to the right angle.
We will use $\sin \left( \alpha \right)=\dfrac{opposite\,side}{hypotenuse}$.

Complete step by step solution:
We have been given that, in a right-angled triangle. The length of side opposite to ${{30}^{\circ }}$angle is of 12.5m. We need to find the length of hypotenuse.

In the above right-angled triangle, side AC is the hypotenuse, and AB has length of 12.5m. The measure of angle B and C is 90 degrees and 30 degrees respectively. We know that in a right-angled triangle the $\sin \left( \alpha \right)=\dfrac{opposite\,side}{hypotenuse}$.
Here, the angle whose sine neeeds to be calculated is C, the side opposite to C is AB and hypotenuse is AC. Putting these in the expression for sine, we get
\[\Rightarrow \sin {{30}^{\circ }}=\dfrac{AB}{AC}\]
We know the value of \[\sin {{30}^{\circ }}\] is \[\dfrac{1}{2}\], substituting this and the length sides we get
\[\Rightarrow \dfrac{1}{2}=\dfrac{12.5m}{AC}\]
Solving the above equation, we get
\[\Rightarrow AC=25m\]
Therefore, the length of hypotenuse is 25m.

Note: To solve these types of questions, we should know how to use trigonometric ratios in a right-angled triangle. For other trigonometric ratios like cosine and tangent, we can find their values in right angle triangle as \[\cos \left( \alpha \right)=\dfrac{adjacent\,side}{hypotenuse}\] and \[\tan \left( \alpha \right)=\dfrac{opposite\,side}{adjacent\,side}\].