Finding Coordinates and Diagonal Difference for a Parallelogram

The problem provides the vertices of a parallelogram in order: A(1, 2), B(4, y), C(x, 6), and D(3, 5). We are asked to find the value of AC² – BD².

In a parallelogram, the diagonals bisect each other. This means the midpoint of the diagonal AC is the same as the midpoint of the diagonal BD. We can use the midpoint formula to find the values of x and y.

Determining the Unknown Vertices (x, y)

The midpoint formula for two points $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$ is $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$.

• Midpoint of AC (using A(1, 2) and C(x, 6)):

$\text{Midpoint}_{AC} = \left(\frac{1+x}{2}, \frac{2+6}{2}\right) = \left(\frac{1+x}{2}, \frac{8}{2}\right) = \left(\frac{1+x}{2}, 4\right)$

• Midpoint of BD (using B(4, y) and D(3, 5)):

$\text{Midpoint}_{BD} = \left(\frac{4+3}{2}, \frac{y+5}{2}\right) = \left(\frac{7}{2}, \frac{y+5}{2}\right)$

Since the midpoints are the same, we can equate the corresponding coordinates:

• Equating x-coordinates:

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

$1+x = 7$

$x = 7 - 1 = 6$

• Equating y-coordinates:

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

$4 \times 2 = y+5$

$8 = y+5$

$y = 8 - 5 = 3$

So, the vertices of the parallelogram are A(1, 2), B(4, 3), C(6, 6), and D(3, 5).

Calculating the Squares of the Diagonals

Now we need to find the lengths of the diagonals AC and BD and then square them. The distance formula between two points $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$ is $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$. The square of the distance is simply $(x_2-x_1)^2 + (y_2-y_1)^2$.

• Length of diagonal AC (using A(1, 2) and C(6, 6)):

$AC^2 = (6-1)^2 + (6-2)^2$

$AC^2 = (5)^2 + (4)^2$

$AC^2 = 25 + 16$

$AC^2 = 41$

• Length of diagonal BD (using B(4, 3) and D(3, 5)):

$BD^2 = (3-4)^2 + (5-3)^2$

$BD^2 = (-1)^2 + (2)^2$

$BD^2 = 1 + 4$

$BD^2 = 5$

Calculating AC² – BD²

Finally, we calculate the required value:

$AC^2 - BD^2 = 41 - 5$

$AC^2 - BD^2 = 36$

Thus, the value of AC² – BD² is 36.

Revision Table

Additional Information on Parallelograms

A parallelogram is a quadrilateral with two pairs of parallel sides. It has several important properties:

• Opposite sides are equal in length.
• Opposite angles are equal in measure.
• Consecutive angles are supplementary (add up to 180°).
• Diagonals bisect each other (as used in this problem).

There is also a property relating the side lengths and diagonal lengths of a parallelogram: the sum of the squares of the diagonals is equal to the sum of the squares of the four sides.

$AC^2 + BD^2 = AB^2 + BC^2 + CD^2 + DA^2$

Since opposite sides are equal (AB=CD and BC=DA), this can also be written as:

$AC^2 + BD^2 = 2(AB^2 + BC^2)$

This property could serve as an alternative way to check some aspects of the parallelogram, although it wasn't directly needed to solve AC² – BD² in this case.