Question

Find the length of the side of the hypotenuse of a right-angled triangle whose leg is of length \[4\] cm and the hypotenuse is \[6\] cm.

Answer 1

Hint:The leg represents any of the sides of a right-angled triangle apart from its hypotenuse. Depict the triangle in form of a diagram to understand the sides and then apply Pythagoras’s theorem.

Complete step by step solution:
A right-angled triangle is one whose one of the angles is \[{90^ \circ }\].
The two sides of the triangle that intersect to form the right angle are called the legs and the side opposite to the right angle is called the hypotenuse.
To determine the length of any side of a right-angled triangle, when the lengths of the other two sides are given, the Pythagoras theorem can be used. The theorem states that for any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
The required triangle in the form of a diagram as shown:

To find the length of AB, use the Pythagoras’s theorem:
According to that:
\[{\left( {AB} \right)^2} + {\left( {BC} \right)^2} = {\left( {AC} \right)^2}\]
\[ \Rightarrow {\left( {AB} \right)^2} = {\left( {AC} \right)^2} - {\left( {BC} \right)^2}\]
\[ \Rightarrow \] \[{\left( {AB} \right)^2} = {6^2} - {4^2}\]
\[ \Rightarrow \] \[{\left( {AB} \right)^2} = 36 - 16\]
\[ \Rightarrow \] \[{\left( {AB} \right)^2} = 20\]
\[ \Rightarrow \] \[AB = \sqrt {20} \]
\[ \Rightarrow \] \[AB = \pm 2\sqrt 5 \]
Since length can’t be negative hence, ignore the negative value;
\[\therefore \] \[AC = 2\sqrt 5 \] cm.

Thus, the length of the unknown side of the right-angled triangle is equal to \[2\sqrt 5 \] cm.

Note: For any right-angled triangle, the legs are also called the base and the perpendicular. The hypotenuse always remains constant but the base and the perpendicular might change depending on which angle we consider. The base is the side adjacent to the acute angle we consider or the side that contains both the right angle and the angle we consider. The perpendicular is the angle opposite to the acute angle of consideration. For example in the above triangle with respect to the \[\angle C\], the side \[BC\] is the base but with respect to the \[\angle A\], the side \[AB\] is the base.