Year 9 Maths Core Pythagoras Theorem

Testing Right-Angled Triangles

Q: What is the converse of Pythagoras' theorem?

It says that if a squared plus b squared equals c squared for the sides of a triangle, then the triangle has a right angle, opposite the side c.

Q: How do builders check a corner is square?

They measure along each edge, for example 1.2 m and 1.6 m, and then measure between the two marks. If the corner is square, that distance is exactly 2 m, because 1.2 squared plus 1.6 squared is 4.

Q: Does the test work with decimals?

Yes. Square all three lengths as usual. For 1.5, 2 and 2.5: 2.25 plus 4 is 6.25, and 2.5 squared is 6.25, so that triangle is right-angled.

20 practice questions 1 video lesson Theory + worked examples

Pythagoras' theorem also works in reverse: if the squares of the two shorter sides add up to the square of the longest side, the triangle has a right angle. This NSW Year 9 Core maths subtopic revises that test, the converse of the theorem, from Stage 4 (NESA outcome MA4‑PYT‑C‑01).

You will order the three sides, compare \(a^2 + b^2\) with \(c^2\) and give a reason, test decimal lengths such as \(2.8\), \(4.5\) and \(5.3\), name which angle is the right angle (the one opposite the longest side), and spot a test done on the wrong pair of sides. Builders use the same check on the corner of a concrete slab, and carpenters use it on the diagonal of a window frame. It is part of Year 9 Core Pythagoras Theorem.

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Testing Right-Angled Triangles - Video - Testing to see if a triangle is right angled Watch

Theory

The converse of Pythagoras' theorem tests whether a triangle is right-angled from its three sides. This Year 9 maths page shows the test: square the two shorter sides, add, and compare with the square of the longest side. It also shows how to name the right angle and how builders use the test to check that a corner is square.

Pythagoras' theorem says: in a right-angled triangle, \(c^2 = a^2 + b^2\). The converse runs the other way. If the sides of a triangle fit \(a^2 + b^2 = c^2\), then the triangle is right-angled.

To test a triangle, find the longest side. Square the two shorter sides and add them. Square the longest side on its own. Compare the two results.

If the results are exactly equal, the triangle is right-angled. If they are different, even by a little, it is not.

When the test works, the longest side is the hypotenuse. The right angle is at the vertex opposite the longest side: the one vertex that side does not touch.

Builders, carpenters and fencers use this to check that a corner is square. They measure along both edges, then measure across between the two marks.

Measure \(a\) and \(b\) along the two edges, then measure \(c\) between the marks. If \(a^2 + b^2 = c^2\), the corner is a true right angle.

When the test works, the right angle is at the vertex opposite the longest side, the one vertex the longest side does not touch.

With \(c\) the longest side and \(a\) and \(b\) the other two:

\[a^2 + b^2 = c^2 \;\Rightarrow\; \text{right-angled} \qquad a^2 + b^2 \ne c^2 \;\Rightarrow\; \text{not right-angled}\]

a^{2} + b^{2} = c^{2}

Naming the right angle. It is at the vertex opposite the longest side. In \(\triangle PQR\), if \(QR\) is the longest side and the test works, the right angle is at \(P\): \(\angle QPR = 90^\circ\).

Multiples. If \(3, 4, 5\) works, so does every multiple: \(6, 8, 10\) or \(1.5, 2, 2.5\). Multiplying every side by the same number keeps the triangle right-angled. Changing only one side does not.

A rectangle. Check that a frame is a true rectangle by testing one corner: the length and the width are the shorter sides, and the diagonal is the longest side.

How to test whether a triangle is right-angled

Find the longest side. It is the only possible hypotenuse.
Square the two shorter sides and add them.
Square the longest side on its own.
Compare. Equal: right-angled. Not equal: not right-angled.
Name the right angle, if asked: it is at the vertex opposite the longest side.

Example 1 — A triangle that passes

Is a triangle with sides \(10\) cm, \(24\) cm and \(26\) cm right-angled?

Solution

The longest side is \(26\) cm. Square the two shorter sides and add. Then square the longest side on its own:

\(10^2 + 24^2\)	\(=\)	\(100 + 576\)
\(\)	\(=\)	\(676\)
\(26^2\)	\(=\)	\(676\)

10^{2} + 24^{2} = 26^{2}

The two results are equal, so yes, the triangle is right-angled. The right angle is opposite the \(26\) cm side.

Example 2 — Is the corner square?

A builder marks \(1.6\) m along one edge of a concrete slab and \(1.2\) m along the other. The distance between the two marks is \(2.1\) m. Is the corner a true right angle?

Solution

The three measurements make a triangle. The longest is \(2.1\) m. Test it:

\(1.6^2 + 1.2^2\)	\(=\)	\(2.56 + 1.44\)
\(\)	\(=\)	\(4\)
\(2.1^2\)	\(=\)	\(4.41\)

4 \neq 4.41

\(4 \ne 4.41\), so no, the corner is not a true right angle. Close is not enough: the two results must be exactly equal. (For a square corner the marks would be \(2\) m apart, because \(\sqrt{4} = 2\).)

Example 3 — Decimal side lengths

Is a triangle with sides \(1.5\) m, \(2.0\) m and \(2.5\) m right-angled?

Solution

Decimals change nothing about the test. The longest side is \(2.5\) m:

\(1.5^2 + 2.0^2\)	\(=\)	\(2.25 + 4\)
\(\)	\(=\)	\(6.25\)
\(2.5^2\)	\(=\)	\(6.25\)

{1.5}^{2} + 2^{2} = {2.5}^{2}

Equal, so yes, it is right-angled. Notice the sides are \(3, 4, 5\) each halved. Multiplying or dividing every side of a right-angled triangle by the same number keeps it right-angled.

Example 4 — Which angle is the right angle?

In \(\triangle DEF\), \(DE = 65\), \(EF = 72\) and \(DF = 97\). Show that the triangle is right-angled and name the right angle.

Solution

The longest side is \(DF = 97\). The other two must be the shorter sides:

\(DE^2 + EF^2\)	\(=\)	\(65^2 + 72^2\)
\(\)	\(=\)	\(4225 + 5184\)
\(\)	\(=\)	\(9409\)
\(DF^2\)	\(=\)	\(97^2 = 9409\)

65^{2} + 72^{2} = 97^{2}

They are equal, so the triangle is right-angled. The right angle is opposite the longest side \(DF\). \(DF\) does not touch \(E\), so the right angle is at \(E\): it is \(\boldsymbol{\angle DEF}\).

Common pitfalls

The longest side stands alone. Add the squares of the two shorter sides. Adding the longest side to another one tests the wrong thing and can give the wrong verdict.

Close is not equal. \(4\) and \(4.41\) are close, but the corner is not square. The two results must match exactly.

Do not add the sides themselves. \(10 + 24 \ne 26\) proves nothing. Square first, then add.

A diagram may not be to scale. A triangle can look right-angled and not be one, and a right-angle mark may be missing. Only the numbers decide.

Frequently asked questions

How do you prove a triangle is right-angled?

Use the converse of Pythagoras' theorem. Square the two shorter sides and add. If the total equals the square of the longest side, the triangle is right-angled.

What is the converse of Pythagoras' theorem?

It says that if \(a^2 + b^2 = c^2\) for the sides of a triangle, then the triangle has a right angle, opposite the side \(c\).

Where is the right angle in a right-angled triangle?

It is opposite the longest side, at the vertex the longest side does not touch.

How do builders check a corner is square?

They measure along each edge, for example \(1.2\) m and \(1.6\) m, and then measure between the two marks. If the corner is square, that distance is exactly \(2\) m, because \(1.2^2 + 1.6^2 = 4\).

Does the test work with decimals?

Yes. Square all three lengths as usual. For \(1.5\), \(2\) and \(2.5\): \(2.25 + 4 = 6.25 = 2.5^2\), so that triangle is right-angled.

Testing Right-Angled Triangles

📖 Theory

How to test whether a triangle is right-angled

Common pitfalls

Frequently asked questions

Theory