(Confusing) Sine/Cosine rule for Question Regarding Obtuse Triangle / GCSE

Question

Background: This question was taken from Pearson's Edexcel GCSE (9-1) Mathematics Algebra and Shape Workbook

Question focus: 3(a)

Triangle WXZ - Diagram (link to image, provided by stack exchange)

According to the question (a), I am being told to 'calculate the length of WY'.

Using the sine rule/law, it is possible to find the length as WY = 7.4 cm. This is shown in the answers page.

However, the question can be solved using the cosine rule/law too, and such calculations reveal the answer as WY = 8.12 cm.

How is this so? A subsequent to-scale diagram shows that WY = 8.15 cm, similar to the answer yielded by the cosine rule. Therefore, it appears that the cosine rule derived solution is correct. (Cosine rule utilises sides and angles which form SAS, meaning there is no other triangle which could fit the description provided)

One possibility of error is the ambiguous case of the sine law, where triangles fulfilling the SSA (side-side-angle) criteria may have more than one triangle which meets the description, as SSA does not prove triangles as congruent

HOWEVER, this is not the case as we know WZ = 14 cm; <WZY = 30°; <WYZ = 110° which is AAS, a congruent triangle proof.

Therefore, this question leaves me with no convincing explanation of why different methods yield conflicting solutions (and why Pearson decided to include the erroneous result in the answers page)

An explanation would be kindly received!

Answer 1

The given diagram is impossible. By the cosine rule, we find that \begin{align*} WY^2 &= 14^2 + 8^2 - 2(14)(18)\cos 30^\circ =148 \\ \implies WY &\approx 8.12 \end{align*} But by the sine rule, \begin{align*} \frac{\sin \angle ZYW}{14} &\approx \frac{\sin{30^\circ}}{8.12} \\ \implies \angle ZYW &\approx 180^\circ - \arcsin\left(\frac{14 \sin 30^\circ}{8.23}\right) \\ &\approx 120.5^\circ \ne 110^\circ. \end{align*} After fixing the angle $WZY$ and the length $ZW$, the point $W$ is uniquely specified, and the rest of the diagram is not compatible with this point.

Answer 2

As noted in the comments and @kipf's answer, the problem is that the given measurements are incompatible. Here are accurate diagrams:

• If you assert $|YZ|=8$, then applying the Law of Cosines gives $|WY|=8.12$. However, $\angle Y = 120.51^\circ$, not $110^\circ$.

• If you assert $\angle Y = 110^\circ$, then applying the Law of Sines (unambiguously) gives $|WY|=7.45$. However, $|YZ|=9.58$, not $8$.

So, you get different answers from applying different Laws simply because you're solving different triangles.

Incompatible measurements are not-uncommon in textbook/test diagrams, where there's a desire to present "nice" (typically, integer or rational) measures of sides and angles, even if the resulting triangles are actually impossible. (Fun Fact: With triangles, you can either have "nice" sides or "nice" angles, but not both. This is one way to interpret Niven's Theorem.) Ideally, when this happens, the corresponding exercise would specify which measures to accept, and which Law to apply, "for now", to avoid contradictory answers. (Alternatively, official solutions could include results from both approaches, with the author taking the opportunity to highlight the issue and advise caution.)