Why doesn't the GCSE syllabus allow us to use herons formula?

Question

I had this question in an exam:

In the triangle below, $AB = 12cm$, $BC = 19cm$ and $AC = 14cm$. Calculate the area of the triangle.

The answer to this question finds the angle $A$ using the cosine rule and then uses this formula to find the area:

$$ \frac { 1 }{ 2 } ab \sin { A } $$

Why can't I just use Heron's formula, where the area of $\triangle ABC$ with perimeter $S$ is:

$$ \sqrt { s(s - a) (s - b) (s - c) } $$

This is a GCSE question.

Answer 1

Bit late on this I know.
Heron's formula is fun but often painful and long; most of the time in GCSE it would be much more work. It is also off-syllabus as the GCSE course tries to teach you trigonometry - so there is no point in teaching you shortcuts for trigonometry right now as the cosine rule etc. will lead on to more important and difficult trigonometry in A Level maths, for example.

Answer 2

I'm guessing this is because they are only asking the question to test the candidates ability to notice and apply trigonometry. Most candidates will not have been taught Heron's formula and so they are not expecting anyone to use it. I imagine that any correct answer would be awarded full marks regardless unless they specify which method to use as part of the question.

Answer 3

Following my comment $$2bc\cos A =b^2+c^2-a^2$$

I'll use $X$ for the area, so that $4X=2bc \sin A$

Square these two and add:

$$16X^2+(b^2+c^2-a^2)^2=4b^2c^2$$ so that $$16X^2=4b^2c^2-(b^2+c^2-a^2)^2$$

You can use this directly to compute $X$ - possibly more easily than Heron.

If we factorise the right-hand side as the difference of two squares we get:

$$16X^2=(2bc+b^2+c^2-a^2)(2bc-b^2-c^2+a^2)=\left((a+b)^2-c^2\right)\left(a^2-(b-c)^2\right)$$

Use the difference of two squares on this to get $$16X^2=(a+b+c)(a+b-c)(a-b+c)(-a+b+c)$$ which easily reduces to Heron's formula - but why do the extra work?

Answer 4

Seconding Karl's answer that the exam may not actually disallow Heron's formula (as claimed), due to the British exam boards practising "positive marking" these days:

From a 2018 GCSE mark scheme:

GENERIC MARKING PRINCIPLE 3:

Marks must be awarded positively:

• marks are awarded for correct/valid answers, as defined in the mark scheme. However, credit is given for valid answers which go beyond the scope of the syllabus and mark scheme, referring to your Team Leader as appropriate