A right triangle has one angle of \(90^\circ\). The two sides that form the right angle are called legs, and the longest side — the one opposite the right angle — is called the hypotenuse.

The Pythagorean Theorem: the sum of the squares of the legs equals the square of the hypotenuse:

where \(a\) and \(b\) are the legs and \(c\) is the hypotenuse. It follows that:

• Finding the hypotenuse: \(c = \sqrt{a^2 + b^2}\)
• Finding a leg: \(a = \sqrt{c^2 - b^2}\)

Pythagorean triples are sets of whole numbers that satisfy the theorem. The most common ones are worth memorising:

Note: any multiple of a triple is also a triple. For example, \(6\text{-}8\text{-}10\) is twice the \(3\text{-}4\text{-}5\) triple.

Example 1: Finding the Hypotenuse from Two Legs

Problem: In a right triangle the legs are \(9\) cm and \(40\) cm. Find the length of the hypotenuse.

Solution:

1. Label the legs \(a = 9\), \(b = 40\), and the hypotenuse \(c\).
2. By the Pythagorean Theorem: \(c^2 = a^2 + b^2 = 9^2 + 40^2\).
3. Calculate: \(81 + 1600 = 1681\).
4. Take the square root: \(c = \sqrt{1681} = 41\).
5. Note that this is the Pythagorean triple \(9\text{-}40\text{-}41\), and the hypotenuse is indeed larger than both legs.

Answer: The hypotenuse is \(41\) cm.

Example 2: Finding a Leg Using a Triple

Problem: In a right triangle the hypotenuse is \(26\) cm and one leg is \(10\) cm. Find the other leg.

Solution:

1. Label \(c = 26\) (hypotenuse), \(a = 10\), and the unknown leg \(b\).
2. Isolate: \(b^2 = c^2 - a^2 = 26^2 - 10^2\).
3. Calculate: \(676 - 100 = 576\).
4. Take the square root: \(b = \sqrt{576} = 24\).
5. This is twice the \(5\text{-}12\text{-}13\) triple, giving \(10\text{-}24\text{-}26\).

Answer: The other leg is \(24\) cm.

Example 3: Checking Whether a Triangle Is a Right Triangle

Problem: A triangle has sides \(20\), \(21\), and \(29\). Is it a right triangle?

Solution:

1. The longest side is \(29\), so if the triangle is a right triangle it must be the hypotenuse.
2. Check whether \(20^2 + 21^2 = 29^2\) holds.
3. Left side: \(400 + 441 = 841\).
4. Right side: \(29^2 = 841\).
5. Both sides are equal, so by the converse of the Pythagorean Theorem the triangle is a right triangle.

Answer: Yes, the triangle is a right triangle (the hypotenuse is the side of length \(29\)).

Example 4: Applied Problem — Ladder Leaning Against a Wall

Problem: A \(17\)-metre ladder leans against a vertical wall with its base \(8\) metres from the base of the wall. How high up the wall does the top of the ladder reach?

Solution:

1. The ladder, wall, and ground form a right triangle. The ladder is the hypotenuse \(c = 17\), the ground distance is a leg \(a = 8\), and the height \(b\) is the unknown leg.
2. Write: \(b^2 = c^2 - a^2 = 17^2 - 8^2\).
3. Calculate: \(289 - 64 = 225\).
4. Take the square root: \(b = \sqrt{225} = 15\).
5. This is the \(8\text{-}15\text{-}17\) Pythagorean triple.

Answer: The top of the ladder reaches a height of \(15\) metres.

✗ Common mistake: Adding the squares of all three sides even when looking for a leg, resulting in an overly large hypotenuse.

✓ The correct way: When finding a leg, subtract: \(b^2 = c^2 - a^2\). Addition is used only when the hypotenuse is the missing side. Always confirm which side is the hypotenuse first.

✗ Common mistake: Confusing the hypotenuse with a leg and substituting the hypotenuse in the leg's place in the formula.

✓ The correct way: The hypotenuse is always the side opposite the right angle and is the longest side. Label it \(c\) before you begin calculating.

✗ Common mistake: Forgetting to take the square root at the end and leaving the answer as a squared value — for example, stating that the hypotenuse is 1681 instead of 41.

✓ The correct way: The formula gives the square of the side. The final step is always to take the square root: \(c = \sqrt{c^2}\).

The key formula: \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse.

• Finding the hypotenuse: \(c = \sqrt{a^2 + b^2}\)
• Finding a leg: \(a = \sqrt{c^2 - b^2}\)
• Triples to remember: \(3\text{-}4\text{-}5\), \(5\text{-}12\text{-}13\), \(8\text{-}15\text{-}17\), \(7\text{-}24\text{-}25\) and their multiples.
• Checking for a right triangle: compare \(a^2 + b^2\) with \(c^2\) of the longest side.