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Year 10 Maths | Understand, justify, check

Choose the right relationship before squaring

Pythagoras relates the side lengths of a right-angled triangle. Before substituting numbers, identify the right angle and the side opposite it. That side is the hypotenuse, and its square equals the sum of the squares of the shorter sides.

What to notice

The hypotenuse is opposite the right angle and is the longest side. A diagram’s orientation does not determine which symbol belongs in the c position of a² + b² = c².

What to do

Identify whether the unknown is the hypotenuse or a shorter side. Add the two smaller squares to find the hypotenuse squared, or subtract a known smaller square from the hypotenuse squared to find another side.

Why it works

The equation relates areas of squares on the sides. Taking a square root returns a length. Keeping these stages separate prevents an area-like result from being reported as a side length.

Work through an example

A right triangle has shorter sides 3 cm and 4 cm. Find its hypotenuse.

3 cm4 cmc
Right triangle with perpendicular sides 3 cm and 4 cm; the opposite side is c.
  1. c² = 3² + 4² = 9 + 16 = 25.
  2. Take the positive square root: c = 5 cm.
  3. 5 cm is longer than either shorter side, which is a useful check.

A common mistake

Adding side lengths is not Pythagoras. Nor is the sum of their squares the final length: a square root is still required.

A strategy to try

When testing three side lengths, square the longest side separately. Compare it with the sum of the other two squares before claiming a right angle.

Try it now

Try three questions

No account is needed. These examples do not change saved learning progress. Choose an answer and check it for an explanation; you can try again.

1. A right triangle has perpendicular sides 6 cm and 8 cm. What is its hypotenuse?
6 cm8 cmc
Right triangle with perpendicular sides 6 cm and 8 cm; the hypotenuse is c.
Help: a starting point

Identify the side opposite the marked right angle.

Help: the next step

Add the squares of the perpendicular sides, then take the positive square root.

2. A right triangle has hypotenuse 13 cm and one shorter side 5 cm. Find the other side.
Help: a starting point

Place the hypotenuse on the side of the equation containing a single square.

Help: the next step

Subtract the known shorter-side square before taking the square root.

3. A triangle has sides 6 cm, 8 cm and 11 cm. Which statement correctly tests whether it is right-angled?
Help: a starting point

Use the longest side as the candidate hypotenuse.

Help: the next step

Compare its square with the sum of the squares of the other two sides.

For parents and carers

Ask the learner to point to the hypotenuse before choosing an operation. The examples use exact integer lengths so attention stays on the relationship. They do not require trigonometry, and the diagrams support reasoning rather than measurement with a ruler.

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