Higher June 2025 Paper 3 Q8
8 The diagram shows trapezium \(ABCD\).
\(AB = 12\) cm
\(AD = 8\) cm
The trapezium has an area of 112 cm2
Work out the perimeter of the trapezium.
Give your answer correct to 3 significant figures. (5)
| Answer | Mark | Mark scheme |
|---|---|---|
| 44.9 | P1 | for process to find an expression for the area of the trapezium, eg \(\tfrac{1}{2}(12 + CD)8\) or \(8 \times 12 + \tfrac{1}{2} \times 8 \times x\) or for process to find the area of the triangle, eg \(112 - 8 \times 12\ (= 16)\) |
| P1 | for forming an equation and isolating terms in the unknown length, eg \(4CD = 112 - 48\) or \(\tfrac{1}{2} \times 8 \times x = 112 - 8 \times 12\) or \(\tfrac{1}{2} \times 8 \times x = \text{``}16\text{''}\) or \(CD = 16\) or \(x = 4\) | |
| P1 | for start of process to find length of \(BC\), eg \(8^2 + \text{``}4\text{''}^2\ (= 80)\) or \(8^2 + [\text{their } x]^2\) or \(\tan^{-1}\left(\dfrac{\text{``}4\text{''}}{8}\right)\ (= 26.5\ldots)\) oe or \(\tan^{-1}\left(\dfrac{8}{\text{``}4\text{''}}\right)\ (= 63.4\ldots)\) where “4” can be [their \(x\)] | |
| P1 | for \(\sqrt{8^2 + \text{``}4\text{''}^2}\) or \(\sqrt{64 + \text{``}16\text{''}}\) or \(\sqrt{80}\) or \(4\sqrt{5}\ (= 8.9\ldots)\) oe or \(\sqrt{8^2 + [\text{their } x]^2}\) or \(\dfrac{\text{``}4\text{''}}{\sin \text{``}26.5\ldots\text{''}}\) or \(\dfrac{8}{\cos \text{``}26.5\ldots\text{''}}\) or \(\dfrac{8}{\sin \text{``}63.4\ldots\text{''}}\) or \(\dfrac{\text{``}4\text{''}}{\cos \text{``}63.4\ldots\text{''}}\) where “4” can be [their \(x\)] | |
| A1 | for answer in the range 44.9 to 44.95 |
Additional guidance
\(x\) is the length of the line from \(C\) to where perpendicular from \(B\) meets \(CD\)
Allow use of other letters in place of \(CD\) and \(x\) (do not have to be defined unless otherwise stated)
Award P1 for \(112 - 8 \times 12\ (= 16)\) even if not used
Award P2 for \(CD = 16\) or \(x = 4\) even if not used unless clearly from incorrect working eg \(12 - 8\ (= 4)\)
Only award P2 for 16 if it is clearly identified as \(CD\)
[their \(x\)] can be any value less than 12 or clearly identified as the length of the line from \(C\) to where perpendicular from \(B\) meets \(CD\) (may be seen on the diagram)
Award P4 for (\(BC =\)) \(\sqrt{80}\) or \(4\sqrt{5}\) or 8.9… unless \(x = 4\) is clearly from incorrect working
If an answer is shown in the range in working and then incorrectly rounded award full marks