The Pythagorean Theorem · Grade 8
How do you check whether a triangle is right-angled?
Test the three sides using the converse: if the squares of the two shorter sides add up to exactly the square of the longest, the triangle is right-angled, and the right angle sits opposite that longest side. If the sum is larger, that angle is acute; if smaller, it is obtuse.
Learning objectives
- State the converse of the theorem and explain how it differs from the theorem itself
- Test three given lengths and decide whether they form a right triangle
- Use the converse to prove that a particular angle in a figure is a right angle
Two theorems, two directions
The theorem itself starts from a right angle and concludes something about the sides. The converse starts from the sides and concludes that the angle is right — the same equation, with the reasoning running the other way.
This distinction is not a matter of phrasing. What the question gives you decides: told the triangle is right-angled, you calculate a side; given three sides, you test whether it is right-angled. Each needs the other direction.
How to run the test
Identify the longest side — it is the only candidate for the hypotenuse. Square the two shorter ones, add them, and compare with the square of the longest.
Exact equality means a right angle. There is no 'almost' here: if one side is larger by even a little, the triangle is not right-angled, only close to it.
What an inequality tells you
If the sum of the squares of the shorter sides is larger than the square of the longest, the angle opposite the longest side is less than ninety degrees, which is to say acute. If it is smaller, that angle is obtuse.
This is useful beyond the test itself. It lets you say something about the shape of a triangle from lengths alone, which is why the converse shows up in proof questions and not only in identification ones.
In a diagram proof, the converse is usually the short route. Instead of measuring an angle you measure three lengths and demonstrate a single equality — and that is a complete proof.
Worked examples
Is a triangle with sides 9, 12 and 15 right-angled?
- The longest is 15, so it is the candidate hypotenuse
- Square the shorter two: 81 and 144
- Add: 225
- Square the longest: 15 squared is 225 — equal
Answer: Yes. It is a 3-4-5 triangle scaled by 3
Is a triangle with sides 4, 5 and 6 right-angled?
- The longest is 6
- Sum of the shorter squares: 16 plus 25 is 41
- Square of the longest: 36
- 41 is greater than 36, so there is no equality
Answer: No. The sum is larger, so the angle opposite 6 is acute
Is a triangle with sides 5, 6 and 9 right-angled?
- The longest is 9
- Sum of the shorter squares: 25 plus 36 is 61
- Square of the longest: 81
- 61 is less than 81
Answer: No. The sum is smaller, so the angle opposite 9 is obtuse
Common mistakes
- Testing with a side that is not the longest as the hypotenuse
- Only the longest side can be the hypotenuse. Testing with another will never produce equality, and the conclusion that the triangle is not right-angled would be wrong.
- Accepting 'very close'
- The converse requires exact equality. A difference of one between the two sides means the triangle is not right-angled, however much the drawing suggests otherwise.
- Using the theorem itself when asked whether an angle is right
- The theorem assumes the right angle to begin with, so it cannot prove it. Using it here assumes what is to be shown, and the proof will not be accepted.
What to remember
- Three sides given — use the converse.
- Only the longest side can be the hypotenuse.
- Exact equality means a right angle.
- Larger sum means acute; smaller means obtuse.