Using Pythagoras' theorem within a two-stage calculation to find a length, and applying the properties of shapes and angles to determine an angle in a calculation involving at least two steps.

What this dot point is asking

The SQA wants you to use Pythagoras' theorem within a two-stage calculation to find a length, and to apply the properties of shapes and angles to find an angle in a calculation involving at least two steps, explaining the reasoning.

Pythagoras' theorem

Pythagoras' theorem relates the three sides of a right-angled triangle. The hypotenuse is the longest side, opposite the right angle. The theorem only works for right-angled triangles, so the first thing to check is that the triangle has a right angle; if it does not, Pythagoras cannot be used. In an applied problem the right angle is often where a wall meets the ground, or where a vertical post meets a horizontal base, so look for it before setting up the calculation.

A two-stage Pythagoras problem uses the length you find in a further calculation, such as a perimeter, a second triangle, or doubling for two identical braces. A common second stage is to find a length using Pythagoras and then use it inside a shape: for example, finding the slant height of a roof, then using it to work out the area of the sloping face. Always keep the unrounded length for the second stage and round only the final answer.

Angle properties over two steps

Angle problems use the properties of shapes and lines, usually combining two facts to reach the answer.

Examples in context

These skills solve practical geometry: the diagonal of a gate or screen (Pythagoras), the distance a ladder stands from a wall, the bracing in a frame, and the angles in a roof truss or a tiled pattern. Each rests on the right-angled-triangle relationship or a chain of angle facts, used over more than one step, the skills here.

Pythagoras can also test whether a corner is a true right angle: if the three sides satisfy $c^2 = a^2 + b^2$, the angle opposite the longest side is exactly $90^\circ$. Builders use this to check that a frame or foundation is square, which is another way the two-stage reasoning appears in context.

Try this

Q1. Find the hypotenuse of a right-angled triangle with sides $9$ cm and $12$ cm. [2 marks]

• Cue. $\sqrt{9^2 + 12^2} = \sqrt{225} = 15$ cm.

Q2. A right-angled triangle has hypotenuse $13$ cm and one side $5$ cm. Find the other side. [2 marks]

• Cue. $\sqrt{13^2 - 5^2} = \sqrt{144} = 12$ cm.

Q3. Two angles of a triangle are $55^\circ$ and $65^\circ$. Find the third. [2 marks]

• Cue. $180 - 55 - 65 = 60^\circ$.