Sheet № 108 · Foundation + Higher · AQA · Edexcel · OCR
Finding a Shorter Side (Pythagoras) –
Finding a shorter side is the second key skill in Pythagoras' theorem questions. Instead of adding the squares of the two shorter sides, you subtract — and getting the direction of the subtraction right is where many students lose marks.
§Key definitions
Question:
A right-angled triangle has a hypotenuse of 13 cm and one side of 5 cm. Find the other side.
Answer:
a = 12 cm. Check: 5² + 12² = 25 + 144 = 169 = 13². Correct.
Q1 (Foundation):
A right-angled triangle has a hypotenuse of 17 cm and one side of 8 cm. Find the other side.
Q2 (Foundation):
A rope stretches from the top of a 12 m pole to a point on the ground 9 m from the base. How long is the rope?
Q3 (Higher):
A right-angled triangle has a hypotenuse of 25 cm and one side of 7 cm. Find the other side.
§Formulas to memorise
a² = c² − b²
a = √(c² − b²)
Rearrange the formula: a² = c² − b².
Check: does a² + b² = c²?
Worked example
A right-angled triangle has a hypotenuse of 13 cm and one side of 5 cm. Find the other side.
Working: a² = c² − b² a² = 13² − 5² a² = 169 − 25 a² = 144 a = √144
⚠ Common mistakes
- ✗Subtracting the wrong way round. Always subtract the smaller square from the larger: a² = c² − b². If you do b² − c² you get a negative, which is impossible.
- ✗Adding instead of subtracting. When finding a shorter side, you subtract. When finding the hypotenuse, you add. Mixing these up is the most common Pythagoras error.
- ✗Misidentifying the hypotenuse. If the question gives you two sides and does not specify which is the hypotenuse, the hypotenuse is always the longest one. If you label a shorter side as the hypotenuse, the subtraction may give a wrong result.
✦ Exam tips
- →Write the rearranged formula a² = c² − b² before substituting — this earns a method mark.
- →Always check your answer: substitute back into a² + b² and confirm it equals c².
- →If the answer is not a whole number, the question will tell you how to round. Keep the full calculator value until the final step.