Moderate -0.8 This is a straightforward application of Heron's formula (or cosine rule followed by area formula) with given side lengths. It requires recall of the appropriate formula and careful arithmetic, but involves no problem-solving insight or conceptual difficulty beyond standard AS-level technique.
10 In this question you must show detailed reasoning.
The diagram shows triangle ABC , where \(\mathrm { AB } = 3.9 \mathrm {~cm} , \mathrm { BC } = 4.5 \mathrm {~cm}\) and \(\mathrm { AC } = 3.5 \mathrm {~cm}\).
Determine the area of triangle ABC .
Or \(\cos B = \frac{4.5^2+3.9^2-3.5^2}{2\times3.9\times4.5}\) or \(\cos C = \frac{3.5^2+4.5^2-3.9^2}{2\times4.5\times3.5}\). Correct use of cosine rule; can be in form \(a^2 = b^2 + c^2 - 2bc\cos A\)
\(\cos A = 0.2641\ldots\) correct to 2 or more sf
A1
\(\cos B = 0.66125\ldots\) or \(\cos C = 0.5488889\)
Or \(\frac{1}{2}\times4.5\times3.9\times\sin'48.604'\) or \(\frac{1}{2}\times3.5\times4.5\times\sin'56.709'\). Must use included angle for the two adjacent sides. Alternative: find altitude \(h\), e.g. \(h = 3.5\sin C = 2.9256\ldots\) then \(\frac{1}{2}(4.5)(2.9256\ldots) = 6.58267\ldots\)
awrt \(6.58\) or \(6.6\)
A1
## Question 10:
$\cos A = \frac{3.5^2 + 3.9^2 - 4.5^2}{2 \times 3.9 \times 3.5}$ oe | **M1** | Or $\cos B = \frac{4.5^2+3.9^2-3.5^2}{2\times3.9\times4.5}$ or $\cos C = \frac{3.5^2+4.5^2-3.9^2}{2\times4.5\times3.5}$. Correct use of cosine rule; can be in form $a^2 = b^2 + c^2 - 2bc\cos A$
$\cos A = 0.2641\ldots$ correct to 2 or more sf | **A1** | $\cos B = 0.66125\ldots$ or $\cos C = 0.5488889$
$A = 74.686°$ correct to 2 or more sf | **A1** | $B = 48.604°$ or $C = 56.709°$
$\frac{1}{2} \times 3.5 \times 3.9 \times \sin 74.686$ | **M1FT** | Or $\frac{1}{2}\times4.5\times3.9\times\sin'48.604'$ or $\frac{1}{2}\times3.5\times4.5\times\sin'56.709'$. Must use included angle for the two adjacent sides. Alternative: find altitude $h$, e.g. $h = 3.5\sin C = 2.9256\ldots$ then $\frac{1}{2}(4.5)(2.9256\ldots) = 6.58267\ldots$
awrt $6.58$ or $6.6$ | **A1** |
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10 In this question you must show detailed reasoning.\\
The diagram shows triangle ABC , where $\mathrm { AB } = 3.9 \mathrm {~cm} , \mathrm { BC } = 4.5 \mathrm {~cm}$ and $\mathrm { AC } = 3.5 \mathrm {~cm}$.
Determine the area of triangle ABC .
\hfill \mbox{\textit{OCR MEI AS Paper 2 2023 Q10 [5]}}