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Identifying the Hypotenuse

Every time you walk diagonally across a rectangular park to save time, you are using the mathematics behind Pythagoras' theorem. This theorem only applies to a right-angled triangle and cannot be used if the triangle lacks a $90^\circ$ angle. It states that the square of the longest side is equal to the sum of the squares of the two shorter sides.

To use the theorem, you must always identify the hypotenuse first. It is always the longest side of the triangle and sits directly opposite the $90^\circ$ angle. You can easily find it using the "opposite rule": draw an arrow pointing out from the square right-angle symbol, and it will always point directly to the hypotenuse.

Edexcel questions frequently hide right-angled triangles inside other shapes. For example, drawing a diagonal line across a rectangle creates two right-angled triangles where the diagonal is the hypotenuse.

Calculating a Missing Hypotenuse

Finding the longest side of a triangle is as simple as following the SAS mnemonic: Square, Add, Square root. Once you identify your sides, you perform substitution to replace the letters in the formula with the numbers from the question. Because lengths must be positive, you only use the positive square root for your final answer.

Triangle $ABC$ has a right angle at $B$ . $AB = 9\text{ cm}$ and $BC = 12\text{ cm}$ . Calculate $AC$ .

Step 1: Identify the sides. $AC$ is the hypotenuse ( $c$ ) because it is opposite the right angle at $B$ . Let $a = 9\text{ cm}$ and $b = 12\text{ cm}$ .

Step 2: State the formula.

a^2 + b^2 = c^2

Step 3: Substitute the values.

(9\text{ cm})^2 + (12\text{ cm})^2 = c^2

Step 4: Square the values.

81\text{ cm}^2 + 144\text{ cm}^2 = c^2

Step 5: Add the results.

225\text{ cm}^2 = c^2

Step 6: Square root to find the length.

c = \sqrt{225\text{ cm}^2}

Step 7: Final answer. $15\text{ cm}$

Calculating a Missing Shorter Side

You can add to find the longest side, but you must work backwards to find a shorter side. If the question gives you the hypotenuse and one leg, you must subtract the square of the known shorter side from the square of the hypotenuse. A useful logical check is that your calculated shorter side must be smaller than the hypotenuse.

Find side $x$ where the hypotenuse is $15\text{ cm}$ and one shorter side is $9\text{ cm}$ .

Step 1: Identify the sides. $c = 15\text{ cm}$ , $b = 9\text{ cm}$ . We are finding the missing shorter side ( $a = x$ ).

Step 2: Rearrange and state the formula.

a^2 = c^2 - b^2

Step 3: Substitute the values.

x^2 = (15\text{ cm})^2 - (9\text{ cm})^2

Step 4: Square the values.

x^2 = 225\text{ cm}^2 - 81\text{ cm}^2

Step 5: Subtract the known side.

x^2 = 144\text{ cm}^2

Step 6: Square root to find the length.

x = \sqrt{144\text{ cm}^2}

Step 7: Final answer. $12\text{ cm}$

Exam Skills: Units, Surds, and Accuracy

Understanding how to format your final answer explains why some students lose marks despite performing perfect mathematics. Edexcel mark schemes strictly require consistent units; if one side is in centimetres and another in metres, you must convert them to match before substituting them into the formula.

On Paper 1 (Non-Calculator), you will often be required to leave your answer in exact surd form instead of a decimal. For calculator papers, examiners recommend keeping intermediate values exact using the "Ans" memory button on your calculator. You should only round your very final answer, typically to 3 significant figures or 1 decimal place unless the question specifies otherwise.

Students often add the squares when trying to find a shorter side; remember you must subtract the square of the known shorter side from the square of the hypotenuse to avoid a 'Math Error'.

In Edexcel questions labelled 'Diagram NOT accurately drawn', you cannot rely on measuring with a ruler to find the hypotenuse; you must locate the right-angle symbol and look directly opposite it.

Always check that your calculated hypotenuse is the longest side; if it is shorter than either of the other two sides, you have likely subtracted instead of added.

If a question says 'You must state the units in your answer', failing to write them (e.g., cm, m) will cost you the final accuracy (A1) mark, even if the number is perfect.

Avoid premature approximation by keeping intermediate values as exact surds or using your calculator's memory, only rounding at the very end of the calculation.

The fundamental relation in Euclidean geometry stating that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

A triangle containing one angle of exactly 90 degrees.

The longest side of a right-angled triangle, positioned directly opposite the 90-degree angle.

The inverse operation of squaring; the final mathematical step required to find the actual side length from the squared result.

The process of replacing the variables in a formula with specific numerical values provided in a problem.

One of the two sides (also known as legs) that meet at the 90-degree right angle in a triangle.

Ensuring all measurements use the same unit (e.g., all in cm) so the numerical calculation is valid.

An exact numerical answer left under a square root sign rather than being calculated as a rounded decimal.