Question

Find the unknown length x in the given figure.

Answer 1

Hint: In this problem, we are given a right-angled triangle having base length 4 cm and perpendicular height 3 cm. By using the Pythagoras theorem which states that in a right-angle triangle, the sum of square of base and perpendicular is equal to the square of hypotenuse, we can easily obtain the length of hypotenuse.

Complete step-by-step solution -

According to the figure in the problem, we are given a right-angled triangle. The base and perpendicular height of the right-angled triangle is 4 cm and 3 cm respectively. We are required to find the length of hypotenuse. As per the Pythagoras theorem, the sum of square of base and perpendicular is equal to the square of hypotenuse. This can be mathematically expressed as: ${{h}^{2}}={{b}^{2}}+{{p}^{2}}\ldots (1)$
Now, putting h = 3 cm and b = 4 cm in the above equation (1), we get
$\begin{align}
  & {{h}^{2}}={{4}^{2}}+{{3}^{2}} \\
 & {{h}^{2}}=16+9 \\
 & {{h}^{2}}=25 \\
 & h=\sqrt{25} \\
 & h=5cm \\
\end{align}$
Therefore, the length of hypotenuse is 5 cm.

Note: This problem can be alternatively solved by letting one of the angles as $\theta $ . By using trigonometric ratio, the value of $\tan \theta =\dfrac{3}{4}$. By using the trigonometric identity, ${{\sec }^{2}}\theta =1+{{\tan }^{2}}\theta $ we can evaluate the value of $\sec \theta $. From the diagram, $\sec \theta =\dfrac{x}{4}$. Hence, the desired length is obtained.

.