Question 6

Parallelogram: A parallelogram is a quadrilateral where opposite sides are parallel and equal. A key property used here: the diagonals of a parallelogram bisect each other — meaning both diagonals cut each other exactly at their midpoints. So the midpoint of diagonal AC equals the midpoint of diagonal BD.

Midpoint Formula: The midpoint of a line segment joining $\text{A}(x_1, y_1)$ and $\text{B}(x_2, y_2)$ is:

$M = \left(\frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2}\right)$

Since both diagonals share the same midpoint, we set their midpoints equal and solve for $x$ and $y$.

The diagonals of a parallelogram bisect each other.

Step 1 — Find midpoint of AC

Let's use the midpoint formula. The coordinates of A are $(\mathbf{1}, \mathbf{2})$. The coordinates of C are $(\mathbf{x}, \mathbf{6})$.

$M_{AC} = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$

$M_{AC} = \left(\frac{1+x}{2}, \frac{2+6}{2}\right)$

$M_{AC} = \left(\frac{1+x}{2}, \frac{8}{2}\right)$

$\boxed{M_{AC} = \left(\frac{1+x}{2}, 4\right)}$

Step 2 — Find midpoint of BD

Let's use the midpoint formula again. The coordinates of B are $(\mathbf{4}, \mathbf{y})$. The coordinates of D are $(\mathbf{3}, \mathbf{5})$.

$M_{BD} = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$

$M_{BD} = \left(\frac{4+3}{2}, \frac{y+5}{2}\right)$

$\boxed{M_{BD} = \left(\frac{7}{2}, \frac{y+5}{2}\right)}$

Step 3 — Equate midpoints and solve for x

The midpoints of the diagonals must be the same. Let's equate the x-coordinates.

$\frac{1+x}{2} = \frac{7}{2}$

$1+x = 7$

$x = 7-1$

$\boxed{x = 6}$

Step 4 — Equate midpoints and solve for y

Now let's equate the y-coordinates.

$\frac{y+5}{2} = 4$

$y+5 = 4 \times 2$

$y+5 = 8$

$y = 8-5$

$\boxed{y = 3}$

Answer

(i) $x = 6$ (ii) $y = 3$