Understanding Parallelogram Properties

A parallelogram is a special type of quadrilateral where opposite sides are parallel. One key property of parallelograms is that their diagonals bisect each other. This means the midpoint of one diagonal is the same point as the midpoint of the other diagonal.

We are given three vertices of a parallelogram: P(2, 4), Q(8, 12), and R(10, 14). Let the fourth vertex be S(x, y).

In a parallelogram PQRS, the diagonals are PR and QS.

Finding Midpoints Using Coordinate Geometry

The midpoint formula helps us find the coordinates of the point exactly halfway between two given points \((x_1, y_1)\) and \((x_2, y_2)\). The formula is:

Midpoint M = \( (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}) \)

Calculating Midpoint of Diagonal PR

Using the coordinates of P(2, 4) and R(10, 14):

Midpoint of PR = \( (\frac{2 + 10}{2}, \frac{4 + 14}{2}) \)

Midpoint of PR = \( (\frac{12}{2}, \frac{18}{2}) \)

Midpoint of PR = \( (6, 9) \)

Calculating Midpoint of Diagonal QS

Using the coordinates of Q(8, 12) and S(x, y):

Midpoint of QS = \( (\frac{8 + x}{2}, \frac{12 + y}{2}) \)

Equating Midpoints to Find Vertex S

Since the diagonals bisect each other, the midpoint of PR must be the same as the midpoint of QS:

Midpoint of PR = Midpoint of QS

\( (6, 9) = (\frac{8 + x}{2}, \frac{12 + y}{2}) \)

Now, we equate the x-coordinates and the y-coordinates separately:

Equating x-coordinates:

\( 6 = \frac{8 + x}{2} \)

Multiply both sides by 2:

\( 12 = 8 + x \)

Subtract 8 from both sides:

\( x = 12 - 8 = 4 \)

Equating y-coordinates:

\( 9 = \frac{12 + y}{2} \)

Multiply both sides by 2:

\( 18 = 12 + y \)

Subtract 12 from both sides:

\( y = 18 - 12 = 6 \)

So, the coordinates of vertex S are (4, 6).

Calculating the Value of (x + y)

The question asks for the value of \( (x + y) \). We found \( x=4 \) and \( y=6 \).

\( x + y = 4 + 6 \)

\( x + y = 10 \)

Therefore, the value of \( (x + y) \) is 10.