WAEC 2018 · Paper 2 · Q4

In the diagram, $PQR$ is an isosceles triangle with $|PQ| = (2y + x)\text{ cm}$ , $|QR| = 4y\text{ cm}$ and $|PR| = (6y - 2x + 1)\text{ cm}$ , where $|PR| = |QR|$ . If the perimeter of the triangle is $28\text{ cm}$ , find the:

(a)
values of $x$ and $y$ ;
Separate values with commas, e.g. 3, −2
(b)
lengths of the sides of the triangle.
Separate values with commas, e.g. 3, −2

Worked solution (try it first)

(a)

The perimeter is 28 cm: $(2y + x) + 4y + (6y - 2x + 1) = 28$ .
Collect like terms: $12y - x + 1 = 28$ , so $12y - x = 27$ (1).
The two equal sides give a second equation: $6y - 2x + 1 = 4y$ , so $2y - 2x = -1$ (2).
From (1), $x = 12y - 27$ .
Substitute into (2): $2y - 2(12y - 27) = -1$ .
So $2y - 24y + 54 = -1$ , $-22y = -55$ and $y = \frac52$ .
Then $x = 12 \times \frac52 - 27 = 3$ .

(b)

$|PQ| = 2y + x = 5 + 3 = 8\text{ cm}$ .
$|QR| = 4y = 10\text{ cm}$ .
$|PR| = 6y - 2x + 1$
$= 15 - 6 + 1$
$= 10\text{ cm}$ .
Check: $8 + 10 + 10 = 28$ ✓.

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