June 2023 Paper 3 Q8
8 A circle with centre A and radius 8 cm and a circle with centre C and radius 12 cm intersect at points B and D.
Quadrilateral ABCD has area \(60\ \mathrm{cm}^2\).
Determine the two possible values for the length AC. [7]
| Scheme | Marks | AO |
|---|---|---|
| Sketch diagram consistent with information in the question | B1 | 2.5 |
| \(60 = 2 \times \dfrac{1}{2} \times 8 \times 12 \times \sin B\) | M1 | 3.1a |
| \(\sin B = \dfrac{5}{8}\) so \(B = 38.7^\circ\) (0.675 rads) | A1 | 1.1a |
| OR \(B = 141.3^\circ\) (2.47 rads) | A1 | 3.2a |
| \(\mathrm{AC}^2 = 8^2 + 12^2 - 2 \times 8 \times 12\cos 38.7\) \(\qquad = 58.1\) | M1 | 3.1a |
| AC = 7.62 cm | A1 | 1.1 |
| \(\mathrm{AC}^2 = 8^2 + 12^2 - 2 \times 8 \times 12\cos 141.3\) \(\qquad = 357.9\) AC = 18.9 cm | A1 | 1.1 |
| [7] |
Notes
B1: Triangle ADC or ABC or quadrilateral ABCD and 8 and 12 indicated eg side lengths or radii. Circles may or may not be shown.
M1: M1 implies previous B1
These next 3 marks can be for angles B or D.
A1: One value of B or cos B
\(\cos B = \dfrac{\sqrt{39}}{8}\)
A1: Other value of B or cos B
\(\cos B = -\dfrac{\sqrt{39}}{8}\)
M1: Use of cosine rule
A1: Accept 7.6 www
A1: Accept 19 www
A0 if more than 2 answers