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Find a side of a right-angled triangle

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Which conditions allow a ladder, wall and ground to form a right triangle?

Year 10, Chapter 9, 4 difficulty levels, Calculator allowedBuild a worksheet

Find a shorter side

Use the side opposite the right angle as the hypotenuse c. Subtract the square of the known leg b to find the square of the missing leg a.

\[a=\sqrt{c^2-b^2}\]
  1. Identify the sides The hypotenuse is longest. Either leg can be the unknown
  2. Rearrange first Subtract the known leg square from both sides \(\displaystyle a^2=c^2-b^2\)
  3. Match units Convert the lengths before squaring
  4. Check the result The missing leg is positive and less than the hypotenuse. Round only the final result

Worked example

Find the other shorter side of this right-angled triangle in metres. The hypotenuse is labelled c and the known shorter side is labelled b.

\[\begin{aligned}c&=5.8\text{ m}\\b&=400\text{ cm}\end{aligned}\]
  1. \(\displaystyle 400\text{ cm}=4\text{ m}\)Express both known lengths in metres
  2. \(\displaystyle a^2+4^2=5.8^2\)The sum of the squares of the shorter sides equals the square of the hypotenuse
  3. \(\displaystyle a^2+16=33.64\)Evaluate the two known squares
  4. \(\displaystyle a^2+16-16=33.64-16\)Subtract the square of the known shorter side from both sides
  5. \(\displaystyle a^2=17.64\)Simplify both sides
  6. \(\displaystyle a=\sqrt{17.64}\)Take the positive square root to obtain a length
  7. \(\displaystyle a=4.2\text{ m}\)This length is exact
  8. \(\displaystyle 4.2\text{ m}<5.8\text{ m}\)The missing shorter side is shorter than the hypotenuse

Common mistakes

Adding squares for a shorter side
✕\(\displaystyle a^2=c^2+b^2\)✓\(\displaystyle a^2=c^2-b^2\)
Subtract the square of the known shorter side from the square of the hypotenuse
Subtracting lengths
✕\(\displaystyle a=c-b\)✓\(\displaystyle a=\sqrt{c^2-b^2}\)
The relationship is between square areas, not a sum of lengths
Stopping at a squared length
✕\(\displaystyle \begin{aligned}x^2&=185\\x&=185\text{ cm}\end{aligned}\)✓\(\displaystyle \begin{aligned}x^2&=185\\x&=\sqrt{185}\text{ cm}\end{aligned}\)
Take the positive square root to change the squared length into a length

Difficulty levels

introFind a shorter side with a whole-number answer
standardFind a shorter side and round as requested
challengeConvert mixed units before subtracting squares
examChoose a rounded shorter-side length

Questions about this topic

How do I find the hypotenuse?+

Find the right angle. The side opposite it is the hypotenuse, whatever way the triangle is drawn

When do I subtract?+

Subtract when the hypotenuse is known and a shorter side is missing. Subtract the square of the known shorter side from the square of the hypotenuse

When should I round?+

Round the final length only when requested. Keep the exact square root until that step

Must the unknown side be the smallest side?+

No. Both legs are shorter than the hypotenuse, but either leg can be larger than the other

Why is the square-root input negative?+

Check the side roles and units. In a valid right triangle, the hypotenuse exceeds either leg, so c² − b² is positive

What happens if the known leg equals the hypotenuse?+

The other length would be zero. That is a degenerate shape, not a right triangle

Can I use a calculator?+

A calculator is allowed unless a question is marked no 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