Is the hypotenuse the longest side of a right triangle? Justify your answer.

Question

Homi · Accepted Answer

We will use the Pythagorean theorem to compare the lengths of the sides in a right triangle.

Step 1 — Understanding the Pythagorean Theorem

A right triangle has one angle that measures 90 degrees. The side opposite this 90-degree angle is called the hypotenuse. It is usually the longest side. The other two sides are called legs.

Let 'a' and 'b' be the lengths of the two legs. Let 'c' be the length of the hypotenuse. The Pythagorean theorem states a special relationship between these sides.

$a^2 + b^2 = c^2$

Diagram 1

Step 2 — Comparing Side Lengths

We know from the Pythagorean theorem that the square of the hypotenuse equals the sum of the squares of the other two sides.

$c^2 = a^2 + b^2$

Since 'a' and 'b' are lengths, they must be positive numbers. This means $a^2$ and $b^2$ are also positive numbers. So, $c^2$ is equal to $a^2$ plus a positive number ( $b^2$ ). This means $c^2$ must be greater than $a^2$ .

$c^2 > a^2$

Similarly, $c^2$ is equal to $b^2$ plus a positive number ( $a^2$ ). So, $c^2$ must be greater than $b^2$ .

$c^2 > b^2$

If the square of a positive number is greater than the square of another positive number, then the first number itself is greater. Therefore, if $c^2 > a^2$ , then 'c' must be greater than 'a'.

$c > a$

And if $c^2 > b^2$ , then 'c' must be greater than 'b'.

$c > b$

This shows that the side 'c' (the hypotenuse) is longer than both side 'a' and side 'b'.

Answer

(i) $c^2 = a^2 + b^2$ (ii) Therefore, $c^2 > a^2$ and $c^2 > b^2$ (iii) So, $c > a$ and $c > b$ . Hence, the hypotenuse is the longest side of the right triangle.

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