Pythagoras’ theorem
Section: Trigonometry | Syllabus: Cambridge IGCSE Mathematics (0580)
Finding the Hypotenuse
Pythagoras' theorem relates the three sides of a right-angled triangle. It states that the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides.
- Pythagoras' theorem: a^2+b^2=c^2, where c is the hypotenuse and a, b are the two shorter sides
- The hypotenuse is always the longest side, and is always opposite the right angle
- To find the hypotenuse, square both shorter sides, add them together, then take the square root
Worked Example: Finding the Hypotenuse
- Question: A right-angled triangle has shorter sides of 9 cm and 12 cm. Find the length of the hypotenuse.
- Step 1: c^2=9^2+12^2=81+144=225
- Step 2: c=√(225)=15
- Answer: 15 cm
Common Mistakes
MistakeAdding the two shorter sides directly, without squaring them first
Fixboth shorter sides must be squared before adding - simply adding the lengths gives a completely wrong (too small) result
MistakeForgetting to take the square root at the end, leaving the answer as c² instead of c
Fixafter adding the two squares, the square root must still be taken to find the actual length of the hypotenuse
Finding a Shorter Side
If the hypotenuse and one shorter side are known, Pythagoras' theorem can be rearranged to find the missing shorter side.
- Rearranged form: a^2=c^2-b^2 (subtract the known shorter side's square from the hypotenuse's square)
- Always subtract the smaller square from the larger one - the hypotenuse's square is always the largest
- Take the square root at the end, exactly as when finding the hypotenuse
Worked Example: Finding a Missing Shorter Side
- Question: A right-angled triangle has a hypotenuse of 20 cm and one shorter side of 16 cm. Find the length of the other shorter side.
- Step 1: a^2=20^2-16^2=400-256=144
- Step 2: a=√(144)=12
- Answer: 12 cm
Common Mistakes
MistakeAdding the two given values instead of subtracting, when the hypotenuse is one of them
Fixif the hypotenuse is already known, its square must be reduced (subtracted from) - only add squares when finding the hypotenuse itself
MistakeSubtracting the values the wrong way round, e.g. 16²-20², giving a negative number
Fixalways subtract the smaller square (a shorter side) from the larger square (the hypotenuse), never the reverse
Checking Whether a Triangle is Right-Angled
Pythagoras' theorem also works in reverse: if the three side lengths of a triangle satisfy a² + b² = c², then the triangle must contain a right angle.
- To test a triangle, identify the longest side as the potential hypotenuse
- Calculate the sum of the squares of the two shorter sides, and separately calculate the square of the longest side
- If the two results are equal, the triangle is right-angled; if they are not equal, it is not
Worked Example: Testing a Triangle for a Right Angle
- Question: A triangle has sides 7 cm, 24 cm and 25 cm. Determine whether it is right-angled.
- Step 1: The longest side is 25 cm, so test 25² against 7² + 24²
- Step 2: 25^2=625
- Step 3: 7^2+24^2=49+576=625
- Answer: since both equal 625, the triangle is right-angled
Common Mistakes
MistakeSquaring the wrong side as the "hypotenuse" in the test, e.g. testing one of the shorter sides against the sum of the other two squared
Fixalways identify the longest given side first - only that side can be the hypotenuse in a right-angled triangle
MistakeConcluding a triangle isn't right-angled from a small rounding difference, when the values are meant to be exactly equal
Fixrecompute carefully - if using exact whole-number sides, the two totals should match exactly, not just approximately
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