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

Question

Homi · Accepted Answer

Understand the Question

In a right-angled triangle, the side opposite the $90^\circ$ angle is called the hypotenuse.
By the Pythagorean theorem, the square of the hypotenuse equals the sum of the squares of the two legs: $c^2 = a^2 + b^2$ .
Because side lengths are always positive, $c^2$ is strictly greater than both $a^2$ and $b^2$ , which proves $c > a$ and $c > b$ .

Step 1 · Set Up the Pythagorean Theorem

Let the lengths of the two legs of the right triangle be $a$ and $b$ , and let the length of the hypotenuse be $c$ . Diagram 1

By the Pythagorean theorem: $a^2 + b^2 = c^2$

Step 2 · Compare Side Lengths

From the Pythagorean theorem: $c^2 = a^2 + b^2$

Since side lengths must be positive ( $a > 0$ and $b > 0$ ), their squares $a^2$ and $b^2$ are also positive.

Therefore: $c^2 > a^2$ $c^2 > b^2$

Taking the square root for positive values: $c > a$ $c > b$

Thus, the hypotenuse $c$ is longer than both legs $a$ and $b$ .

Answer

Yes, the hypotenuse is the longest side of a right triangle as $c > a$ and $c > b$ .

Common Mistakes

Omitting the Positivity Condition: Forgetting to state that $a, b, c > 0$ , which is required when concluding $c > a$ from $c^2 > a^2$ .
Incomplete Justification: Stating that the hypotenuse is the longest side without using the Pythagorean theorem ( $c^2 = a^2 + b^2$ ) or the angle-side relationship (side opposite the largest angle $90^\circ$ is largest) to justify the claim.

Question 14