Question

Find the length of the hypotenuse of a right triangle, the other two sides of which measure 9 cm and 12 cm.

Answer 1

- Hint: First of all consider a right triangle ABC, right-angled at B. Now, substitute AB = 9 cm and BC = 12 cm and use Pythagoras theorem which is \[{{\left( AB \right)}^{2}}+{{\left( BC \right)}^{2}}={{\left( AC \right)}^{2}}\] to find the value of AC which is the hypotenuse of the triangle.

Complete step-by-step solution -

In this question, we have to find the length of the hypotenuse of a right triangle whose other sides are equal to 9 cm and 12 cm. Before proceeding with this question, let us see what Pythagoras theorem is.
We use the Pythagoras theorem to find the sides of the right triangle. Basically, it is the relation between three sides of the right triangle. Pythagoras Theorem states that “In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The sides of this triangle are named perpendicular, base, and hypotenuse. The hypotenuse is always the longest side in the right triangle as it is the opposite of the longest angle of the triangle which is \[{{90}^{o}}\]. Let us assume a triangle ABC, right-angled at B.

According to Pythagoras theorem,
\[{{\left( AB \right)}^{2}}+{{\left( BC \right)}^{2}}={{\left( AC \right)}^{2}}\]
Now, let us consider our question. Here, we are given that the two sides are equal to 9 cm and 12 cm. So, let us take AB = 9 cm and BC = 12 cm and AC is the hypotenuse.

According to the Pythagoras theorem,
\[{{\left( AB \right)}^{2}}+{{\left( BC \right)}^{2}}={{\left( AC \right)}^{2}}\]
By substituting the values of AB = 9 cm and BC = 12 cm, we get,
\[{{\left( 9 \right)}^{2}}+{{\left( 12 \right)}^{2}}={{\left( AC \right)}^{2}}\]
\[81+144={{\left( AC \right)}^{2}}\]
\[225={{\left( AC \right)}^{2}}\]
So, we get,
\[AC=\sqrt{225}\]
We know that,
\[15\times 15=225\]
So, we get,
\[AC=\sqrt{{{\left( 15 \right)}^{2}}}\]
\[AC=15cm\]
So, we get the value of the hypotenuse as 15 cm.

Note: Students must note that in a right triangle, hypotenuse remains constant but the value of the perpendicular and base changes according to the angle we consider. Perpendicular is the side opposite to and the base is the side adjacent to the acute angle we consider. For example, if we consider \[\angle C\], perpendicular would be AB and base would be BC, while the converse would be true if we consider \[\angle B\]. So, this must be taken care of.