Mastering Pythagoras' Theorem for Your Mathematics Exam

What the examiner is testing

The examiner assesses your ability to accurately identify the hypotenuse and the two shorter sides (legs) of a right-angled triangle. Marks are awarded for correctly substituting these values into the formula \(a^2 + b^2 = c^2\) and then solving for an unknown side, often requiring a square root calculation.

The method

1. Identify the right angle in the triangle.
2. Label the side opposite the right angle as \(c\) (this is the hypotenuse).
3. Label the other two sides as \(a\) and \(b\) (the legs). It doesn't matter which is which.
4. Write down the Pythagoras' theorem formula: \(a^2 + b^2 = c^2\).
5. Substitute the known lengths of sides \(a\), \(b\), or \(c\) into the formula.
6. Rearrange the equation to solve for the unknown side.
7. Calculate the square root of your result to find the length of the unknown side.

Worked example

Calculate the length of the hypotenuse of a right-angled triangle with legs measuring \(5 \text{ cm}\) and \(12 \text{ cm}\).

$$ \begin{aligned} a &= 5 \text{ cm} \\ b &= 12 \text{ cm} \\ c &= ? \\ \\ a^2 + b^2 &= c^2 \\ 5^2 + 12^2 &= c^2 \\ 25 + 144 &= c^2 \\ 169 &= c^2 \\ c &= \sqrt{169} \\ c &= 13 \text{ cm} \end{aligned} $$

Sanity check: The hypotenuse (13 cm) is the longest side, which makes sense as it's opposite the right angle.

Worked example: a harder one

A ladder of length \(6 \text{ m}\) leans against a vertical wall. The base of the ladder is \(2 \text{ m}\) away from the wall. Calculate the height up the wall that the ladder reaches. Give your answer to 3 significant figures.

Initial thought: Some students might mistakenly think \(6 \text{ m}\) is one of the shorter sides. If we set \(a = 6\) and \(b = 2\), then \(c^2 = 6^2 + 2^2 = 36 + 4 = 40\), so \(c = \sqrt{40} \approx 6.32 \text{ m}\). This would mean the hypotenuse is longer than the ladder itself, which is impossible. The ladder is the longest side.

$$ \begin{aligned} a &= ? \\ b &= 2 \text{ m} \\ c &= 6 \text{ m} \\ \\ a^2 + b^2 &= c^2 \\ a^2 + 2^2 &= 6^2 \\ a^2 + 4 &= 36 \\ a^2 &= 36 - 4 \\ a^2 &= 32 \\ a &= \sqrt{32} \\ a &= 5.65685... \text{ m} \\ a &= 5.66 \text{ m (3 s.f.)} \end{aligned} $$

Practice

1. A right-angled triangle has legs of length \(8 \text{ cm}\) and \(15 \text{ cm}\). Find the length of the hypotenuse.
2. The hypotenuse of a right-angled triangle is \(10 \text{ m}\) long. One of its legs is \(6 \text{ m}\) long. What is the length of the other leg?
3. A rectangular field is \(120 \text{ m}\) long and \(50 \text{ m}\) wide. What is the length of the diagonal path across the field?
4. The diagram shows a right-angled triangle. Calculate the value of \(x\). Give your answer to 1 decimal place.
   $$ \begin{array}{c} \text{hypotenuse} = 17 \text{ cm} \\ \text{one leg} = x \text{ cm} \\ \text{other leg} = (x + 7) \text{ cm} \end{array} $$

Answers:
1. \(17 \text{ cm}\)
2. \(8 \text{ m}\)
3. \(130 \text{ m}\)
4. Working for Q4:
$$ \begin{aligned} a^2 + b^2 &= c^2 \\ x^2 + (x+7)^2 &= 17^2 \\ x^2 + (x^2 + 14x + 49) &= 289 \\ 2x^2 + 14x + 49 &= 289 \\ 2x^2 + 14x - 240 &= 0 \\ x^2 + 7x - 120 &= 0 \\ (x+15)(x-8) &= 0 \\ x &= -15 \text{ or } x = 8 \end{aligned} $$
Since length cannot be negative, \(x = 8 \text{ cm}\).

The three mistakes that lose marks

1. Not identifying the hypotenuse correctly: Often, students mistake one of the legs for the hypotenuse when solving for a shorter side.
   • Wrong answer example: If \(c = 10\) and \(a = 6\), incorrectly calculating \(c^2 = a^2 + b^2 \implies 10^2 = 6^2 + b^2 \implies 100 = 36 + b^2 \implies b^2 = 64 \implies b = 8\). This is correct. The mistake is if they think \(a\) is the hypotenuse, so \(a^2 = b^2 + c^2 \implies 6^2 = b^2 + 10^2 \implies 36 = b^2 + 100 \implies b^2 = -64\), which is impossible.
2. Forgetting to take the square root: After calculating \(c^2\) (or \(a^2\) or \(b^2\)), students sometimes state this squared value as the final answer.
   • Wrong answer example: For legs \(3 \text{ cm}\) and \(4 \text{ cm}\), \(c^2 = 3^2 + 4^2 = 9 + 16 = 25\). The wrong answer would be \(25 \text{ cm}\).
3. Incorrectly rearranging the formula when finding a leg: When finding a shorter side, the formula needs to be rearranged to subtract the known leg's square from the hypotenuse's square.
   • Wrong answer example: Given hypotenuse \(c = 10\) and leg \(a = 6\), if a student calculates \(b^2 = 10^2 + 6^2 = 100 + 36 = 136\), then \(b = \sqrt{136} \approx 11.66\). This is incorrect because \(b\) must be shorter than the hypotenuse. The correct calculation is \(b^2 = 10^2 - 6^2 = 100 - 36 = 64\), so \(b = 8\).

30-second recap

Pythagoras' theorem, \(a^2 + b^2 = c^2\), relates the lengths of the three sides of a right-angled triangle. \(c\) is always the hypotenuse (the longest side, opposite the right angle), and \(a\) and \(b\) are the shorter legs. Remember to take the square root at the end and subtract when finding a leg.