Use the converse of Pythagoras' theorem
The converse of Pythagoras' theorem tests whether three given side lengths make a right-angled triangle. Compare the sum of the squares of the two shorter sides with the square of the longest side
\[a^2+b^2=c^2\]
- Identify the longest side Call the longest side c and use it as the possible hypotenuse \(\displaystyle 8<15<17\quad\Rightarrow\quad c=17\)
- Square the shorter sides Square each shorter length separately, then add the two results \(\displaystyle 8^2+15^2=64+225=289\)
- Square the longest side Find the square of the possible hypotenuse \(\displaystyle 17^2=289\)
- Compare the two results Equal square results show that the triangle is right-angled \(\displaystyle 289=289\)
- Check existence Positive shorter lengths must add to more than the longest length
- Respect measurement precision For exact lengths, compare exact squares. For rounded measurements, a small discrepancy may be caused by rounding
Worked example
A triangle has side lengths 9 cm, 12 cm and 15 cm. Determine whether it is right-angled
\[a^2+b^2\quad\text{and}\quad c^2\]
- \(\displaystyle 9<12<15\)The longest side is 15 cm, so it is the possible hypotenuse
- \(\displaystyle 9^2+12^2=81+144=225\)The sum of the shorter-side squares is 225.
- \(\displaystyle 15^2=225\)The square of the longest side is also 225.
- \(\displaystyle 225=225\)The equal square results show that the triangle is right-angled
Common mistakes
Using a shorter side as the hypotenuse
✕\(\displaystyle \begin{gathered}8^2+17^2=15^2\\(353\ne225)\end{gathered}\)✓\(\displaystyle \begin{gathered}8^2+15^2=17^2\\(289=289)\end{gathered}\)
Only the longest side can be the possible hypotenuse, so use 17 cm as c before comparing squaresUsing the triangle inequality as the test
✕\(\displaystyle \begin{gathered}8+15>17\\\Rightarrow\text{right angle}\end{gathered}\)✓\(\displaystyle \begin{gathered}8^2+15^2=17^2\\\Rightarrow\text{right angle}\end{gathered}\)
The side-sum inequality only confirms that the lengths can form a triangle; the converse test compares squaresSquaring the sum of the shorter sides
✕\(\displaystyle \begin{gathered}(8+15)^2=529\\17^2=289\\529\ne289\end{gathered}\)✓\(\displaystyle \begin{gathered}8^2+15^2\\=64+225\\=289=17^2\end{gathered}\)
Square the two shorter sides separately and then add their squares; do not square their sumDifficulty levels
introCompare squares with the longest side listed last; either verdict is possible
standardIdentify the longest side in varied positions and justify either verdict
challengeAvoid treating the last listed side as the hypotenuse; either verdict is possible
examChoose a verdict with a valid reason
Questions about this topic
What does the converse of Pythagoras' theorem test?+
It tests whether three side lengths form a right-angled triangle by comparing the shorter-side square sum with the square of the longest side
Why is the longest side used as the possible hypotenuse?+
The hypotenuse is opposite the right angle and is longer than either other side, so only the longest given side can be the hypotenuse
What does equality of the two square results show?+
If the sum of the shorter-side squares equals the square of the longest side, the triangle is right-angled
Does being longest make a side the hypotenuse?+
Only after a right angle has been established. Before then, call it the possible hypotenuse
What if the side lengths were rounded?+
Use their stated precision. Squared rounded measurements may differ even when the original triangle is right-angled
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 9: Pythagoras and trigonometry
In this chapter
9A Finding the hypotenuse: Find a side of a right-angled triangleDetermine whether three sides form a right-angled triangle9B Finding a shorter side: Find a side of a right-angled triangle9C Pythagoras' theorem in 2D and 3D: Find a side of a right-angled trianglePythagoras in three dimensionsName sides and state trigonometric ratios9D Trigonometric ratios: Evaluate a trigonometric ratio9D Trigonometric ratios: Choose a trigonometric ratio9E Unknown side in the numerator: Find an unknown side of a right-angled triangle9E Unknown side in the numerator: Solve a practical right-angled-triangle problem9E Unknown side in the numerator: Evaluate a trigonometric ratio9E Unknown side in the numerator: Choose a trigonometric ratio9F Unknown side in the denominator: Find an unknown side of a right-angled triangle9F Unknown side in the denominator: Solve a practical right-angled-triangle problem9F Unknown side in the denominator: Choose a trigonometric ratioFind an angle of a right-angled triangle from two sides9G Finding angles with trigonometry: Solve a practical right-angled-triangle problem9G Finding angles with trigonometry: Evaluate a trigonometric ratio, and find an angle from its valueAngles of elevation and depressionRead and state three-figure bearingsSolve a navigation problem with bearingsTrigonometry in three dimensions