WAEC 2023 · Paper 2 · Q13

(a)
$ABCD$ is a trapezium with $\overline{BC} \parallel \overline{AD}$ . $E$ is a point on $AD$ such that $\overline{BE} \perp \overline{AD}$ . If $\angle BDA = 55^\circ$ , $|AE| = 7\text{ cm}$ , $|BE| = 18\text{ cm}$ and $|BC| = 9\text{ cm}$ , find: (i) $\angle BAE$ ; (ii) the area of $ABCD$ .
Separate values with commas, e.g. 3, −2
(b)
A trader bought goods for GH¢ $x$ and sold them for GH¢ $y$ . If the profit was $25\%$ , find an equation connecting $x$ and $y$ .

Worked solution (try it first)

Triangle $ABE$ is right-angled at $E$ : $\tan\angle BAE = \frac{18}{7}$
$\approx 2.571$ , so $\angle BAE \approx 68.75^\circ$ .

In right-angled triangle $BED$ : $|ED| = \frac{18}{\tan 55^\circ}$
$\approx 12.604$ cm, so $|AD| = 7 + 12.604 = 19.604$ cm.
The parallel sides are $BC = 9$ and $AD = 19.604$ , and the height is $BE = 18$ : area $= \frac12(9 + 19.604) \times 18$
$\approx 257.44\text{ cm}^2$ .

A $25\%$ profit means the selling price is $125\%$ of the cost: $y = 1.25x = \frac54x$ .
So $4y = 5x$ , or $x = \frac45y$ .