Converse of Pythagoras' Theorem (SQA National 5 Maths): Revision Note

Students have tended to lose a lot of marks on these questions in past papers.

In particular, do not start by substituting the three sides into $a^2+b^2=c^2$.

• So in the example above, do not write $5^2+{12}^2={14}^2$

• You will lose a mark for doing this, even if everything else is correct

In exams, converse Pythagoras questions have tended to appear in the context of 'real-life' scenarios.

• Look out for questions where you are asked to determine whether two things are perpendicular

• Be sure to answer in the context of the question

A ladder is leaned against a wall built on horizontal ground as shown in the diagram.

The edge of the ladder, AB, is 7 metres long.

C is at the foot of the wall.

A is 3 metres from C.

B is 6 metres from C.

Determine whether the wall is perpendicular to the ground.

Justify your answer.

Answer:

You need to determine whether or not angle ACB is a right angle

• If it is, then the wall is perpendicular to the ground

• If it is not, then the wall is not perpendicular to the ground

Use the converse of Pythagoras' theorem

• Let $c=7$, because the 7 m side is the longest side of the triangle

• $a$ and $b$ will be 3 and 6 (which is which isn't important)

Calculate $a^2+b^2$

$\begin{array}{rcl}{a}^2+b^2& =& 3^2+6^2\\ & =& 9+36\\ & =& 45\end{array}$

Calculate $c^2$

$c^2=7^2=49$

State that those are not equal

$45\ne 49$

Because they are not equal, the triangle is not a right-angled triangle

• This means angle ACB is not a right angle

• Which means the wall isn't perpendicular to the ground

Write your conclusion, giving a justification and answering in the context of the question

The wall is not perpendicular to the ground, because angle ACB is not a right angle