Testing for Right Angled Triangles
Theory
Three side lengths are enough to decide whether a triangle is right-angled. If the squares of the two shorter sides add to the square of the longest side, \(a^2 + b^2 = c^2\), the triangle is right-angled; if not, it is not. This test is the converse of Pythagoras’ theorem. This Year 8 maths page shows how to use it.
Step 1. Find the longest side. It is the only side that could be the hypotenuse, so call it \(c\).
Step 2. Add the squares of the two shorter sides: \(a^2 + b^2\). Step 3. Square the longest side: \(c^2\). Step 4. Compare. If \(a^2 + b^2 = c^2\), the triangle is right-angled. If they are not equal, it is not.
Name the right angle. The right angle is opposite the longest side, at the vertex the longest side does not touch. If the longest side of \(\triangle DEF\) is \(DF\), the right angle is \(\angle DEF\).
Exactly equal. Two results that are close but different mean the triangle is not right-angled.
Same units. Change every length to one unit before you square. Any kind of number. Decimals and surds are tested the same way, because \((\sqrt{7})^2 = 7\).
Don’t judge by eye. A figure may not be drawn to scale, and two triangles can look alike. The side lengths decide. Builders use this test to check that a corner is square: they measure two sides and the diagonal across the corner.
The converse of Pythagoras’ theorem (c is the longest side):
How to test whether a triangle is right-angled
- Change all sides to the same unit.
- Pick the longest side as \(c\).
- Work out \(a^2 + b^2\) and \(c^2\).
- Compare: equal means right-angled; not equal means not.
- Name the right angle: it is opposite the longest side.
The longest side is \(AC = 61\).
| \(AB^2 + BC^2\) | \(=\) | \(60^2 + 11^2 = 3721\) |
| \(AC^2\) | \(=\) | \(61^2 = 3721\) |
\(AB^2 + BC^2 = AC^2\), so \(\triangle ABC\) is right-angled.
The longest side is \(AC = 71\).
| \(AB^2 + BC^2\) | \(=\) | \(13^2 + 70^2 = 5069\) |
| \(AC^2\) | \(=\) | \(71^2 = 5041\) |
\(5069 \ne 5041\), so \(\triangle ABC\) is not right-angled, even though it looks it.
| \(16^2 + 30^2\) | \(=\) | \(256 + 900 = 1156\) |
| \(34^2\) | \(=\) | \(1156\) |
Equal, so the triangle is right-angled.
(i) Is it right-angled?
(ii) Which angle is \(90^\circ\)?
| \(15^2 + 36^2\) | \(=\) | \(225 + 1296 = 1521\) |
| \(39^2\) | \(=\) | \(1521\) |
(i) Yes
(ii) \(DF\) is the longest side and it does not touch \(E\), so the right angle is \(\angle DEF\).
(i) Sides \(60\) cm, \(0.8\) m, \(1\) m
(ii) Sides \(1.2\) m, \(3.5\) m, \(3.6\) m
(i) Change to centimetres first: \(0.8\) m \(= 80\) cm and \(1\) m \(= 100\) cm.
| \(60^2 + 80^2\) | \(=\) | \(3600 + 6400\) |
| \(\) | \(=\) | \(10\,000 = 100^2\) |
| \(1.2^2 + 3.5^2\) | \(=\) | \(1.44 + 12.25\) |
| \(\) | \(=\) | \(13.69\) |
| \(3.6^2\) | \(=\) | \(12.96\) |
(i) Yes, right-angled
(ii) \(13.69 \ne 12.96\), so not right-angled
\(\sqrt{7} \approx 2.6\), so the longest side is \(4\) cm. \((\sqrt{7})^2 = 7\).
| \((\sqrt{7})^2 + 3^2\) | \(=\) | \(7 + 9 = 16\) |
| \(4^2\) | \(=\) | \(16\) |
Equal, so it is right-angled.
(i) Find \(AC\).
(ii) Is \(\angle ACD\) a right angle?
| \(AC^2\) | \(=\) | \(18^2 + 24^2 = 900\) |
| \(30^2 + 16^2\) | \(=\) | \(900 + 256 = 1156\) |
| \(34^2\) | \(=\) | \(1156\) |
(i) \(AC = \sqrt{900} = 30\)
(ii) Yes: in \(\triangle ACD\), \(AD = 34\) is the longest side and it does not touch \(C\), so \(\angle ACD = 90^\circ\).
(i) How long should the third edge be?
(ii) The built deck measures \(3.3\) m, \(4.4\) m and \(5.4\) m. Is the corner square?
| \(c^2\) | \(=\) | \(3.3^2 + 4.4^2\) |
| \(\) | \(=\) | \(10.89 + 19.36 = 30.25\) |
| \(5.4^2\) | \(=\) | \(29.16\) |
(i) \(c = \sqrt{30.25} = 5.5\) m
(ii) \(30.25 \ne 29.16\), so the corner is not square.
Common pitfalls
Frequently asked questions
How do you test if a triangle is right-angled?
Check whether the squares of the two shorter sides add to the square of the longest side. If \(a^2 + b^2 = c^2\), it is right-angled.
What is the converse of Pythagoras' theorem?
If the sides of a triangle fit \(a^2 + b^2 = c^2\), then the triangle has a right angle opposite side \(c\).
Where is the right angle?
Opposite the longest side, at the vertex that side does not touch.
Can a triangle with decimal sides be right-angled?
Yes. \(1.2^2 + 3.5^2 = 13.69 = 3.7^2\), so sides \(1.2\), \(3.5\) and \(3.7\) make a right angle.
Can I tell from the picture?
No. Diagrams may not be drawn to scale. Only the side lengths decide.