The Pythagorean Theorem · Grade 8
How do you find a leg when the hypotenuse is known?
The same theorem, run backwards: you subtract instead of adding. The missing leg squared equals the hypotenuse squared minus the known leg squared, and then you take the root. The step that decides everything is checking first which of the given sides really is the hypotenuse, because subtracting the wrong way round gives a negative number.
Learning objectives
- Rearrange the theorem to isolate an unknown leg
- Calculate a missing side after deciding whether it is a leg or the hypotenuse
- Give the answer as an exact square root or as a rounded decimal, as the question requires
When to add and when to subtract
The only question that matters is which side is missing. If the hypotenuse is unknown, add the squares of the two legs. If a leg is unknown, subtract the square of the known leg from the square of the hypotenuse.
You can see this straight from the theorem. It is an equation, and moving between addition and subtraction is ordinary rearranging: a term added on one side moves to the other with its sign flipped.
The check that prevents the error
Before subtracting, confirm that the larger number is the hypotenuse. Subtracting the other way leads to a negative square, and a length has no root of a negative number — that is how the mistake announces itself.
After the calculation, check again: the leg you found must be shorter than the hypotenuse. A leg longer than the hypotenuse is a contradiction, and it means the substitution was reversed from the start.
How to present the answer
The value under the root is often not a perfect square. If the question wants an exact value, leave the root sign — that is a complete answer, not a partial one.
If it wants a practical length, round as required. Watch the units: if the data are in metres the answer is in metres, and the squares in the middle are not a unit of length at all.
Worked examples
The hypotenuse is 13 and one leg is 5. What is the other leg?
- The unknown is a leg, so subtract
- Hypotenuse squared: 169. Known leg squared: 25
- Subtract: 169 minus 25 is 144
- Take the root: the root of 144 is 12
Answer: The other leg is 12
A 5 metre ladder leans against a wall with its base 3 metres out. How high does it reach?
- The ladder is the hypotenuse; the distance out and the height are the legs
- Subtract: 25 minus 9 is 16
- Take the root: 4
Answer: The ladder reaches 4 metres up the wall
The hypotenuse is 10 and one leg is 6. What is the other leg?
- Confirm 10 really is the longest side: it is
- Subtract: 100 minus 36 is 64
- Take the root: 8
- Check: 8 is shorter than 10, as required
Answer: The other leg is 8
Common mistakes
- Adding when you should subtract
- Adding produces a longer hypotenuse, so the answer comes out longer than the hypotenuse you were given. If the unknown is a leg, the answer must be smaller than the hypotenuse, and that check catches it at once.
- Subtracting the wrong way round
- Subtract the smaller from the larger — the leg squared from the hypotenuse squared. The other direction gives a negative under the root, a sure sign the sides have been confused.
- Rounding the squares during the calculation
- The squares are exact numbers and so is the difference between them. Rounding before taking the root passes the drift into the answer, so round only the final result.
What to remember
- Missing hypotenuse — add. Missing leg — subtract.
- Confirm which side is the hypotenuse first.
- A leg must come out shorter than the hypotenuse.
- A negative under the root means a reversed substitution.