Foundation January 2023 Paper 1 Q23
23 The diagram shows a triangle \(ABC\) inside a semicircle.
Diagram NOT accurately drawn
\(A\), \(B\) and \(C\) are points on the semicircle.
\(AB\) is the diameter of the semicircle.
Angle \(ACB = 90^\circ\)
Angle \(BAC = 50^\circ\)
\(AC = 18\) cm
Work out the perimeter of the semicircle.
Give your answer correct to 2 significant figures.
(5)
| Scheme | Marks |
|---|---|
\(\cos 50 = \dfrac{18}{(AB)}\) or \(\sin 40 = \dfrac{18}{(AB)}\) or \(\dfrac{(AB)}{\sin 90} = \dfrac{18}{\sin 40}\) | M1 |
\((AB =)\;\dfrac{18}{\cos 50}\;(= 28.0030...)\) oe or 28 or \((AB =)\;\dfrac{18}{\sin 40}\;(= 28.0030...)\) oe or 28 | M1 |
\(\dfrac{1}{2} \times \pi \times \text{“}28.0030...\text{”}\;(= 43.9...)\) oe or 44 \(\pi \times \text{“}28.0030...\text{”}\;(= 87.9...)\) oe or 88 | M1 |
| “28…” + “43.9…” (= 71.9900…) or “28” + “44” | M1 |
| Correct answer scores full marks (unless from obvious incorrect working) Answer: 72 | A1 |
| (5) | |
| (5 marks) |
Notes
M1: for use of \(\pi d\) or \(\dfrac{1}{2}\pi d\) oe
Allow any value of \(AB\) > 18 if M2 not scored
M1: ft from previous M1
Allow their \(d\) + their \(\dfrac{1}{2}\pi d\)
A1: awrt 72
M2 for \((AB =)\sqrt{18^2 + (18\tan 50)^2}\) oe (= 28.0030…) or 28