WAEC 2023 · Paper 2 · Q7

(a)
The sum of three numbers is 81. The second number is twice the first. Given that the third number is 6 more than the second, find the numbers.
Separate values with commas, e.g. 3, −2
(b)
Given the points $P(3, 5)$ and $Q(-5, 7)$ on the Cartesian plane such that $R$ is the midpoint of $\overline{PQ}$ , find the equation of the line that passes through $R$ and is perpendicular to $\overline{PQ}$ .

Worked solution (try it first)

(a)

Let the first number be $x$ .
The second is twice the first, $2x$ , and the third is 6 more than the second, $2x + 6$ .
They add up to 81: $x + 2x + (2x + 6) = 81$ .
So $5x + 6 = 81$ , $5x = 75$ and $x = 15$ .
The numbers are 15, 30 and 36.

(b)

Midpoint $R = \left(\frac{3 + (-5)}{2}, \frac{5 + 7}{2}\right)$
$= (-1, 6)$ .
Gradient of $PQ = \frac{7 - 5}{-5 - 3}$
$= \frac{2}{-8}$
$= -\frac14$ .
A perpendicular line has gradient $m$ with $m \times \left(-\frac14\right) = -1$ , so $m = 4$ .
The line through $R(-1, 6)$ with gradient 4: $y - 6 = 4(x + 1)$ .
So $y = 4x + 10$ .

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