Question

How do you find the midpoint of each diagonal of the quadrilateral with vertices P(1,3), Q(6,5), R(8,0), and S(3,-2)?

Answer 1

Midpoint of the diagonal PR : `color(blue)((4.5, 1.5)`

Midpoint of the diagonal QS : `color(blue)((4.5, 1.5)`

Both the diagonals have the same Midpoint, and we have a Parallelogram.

Explanation:

We are given a Quadrilateral with the following Vertices:

`color(brown)(P(1,3), Q(6, 5), R(8,0), and S(3,-2)`

The MidPoint Formula for a Line Segment with Vertices `color(blue)((x_1, y_1) and (x_2, y_2):`

`color(blue)([(x_1+x_2)/2, (y_1+y_2)/2]`

`color(green)(Step.1`

Consider the Vertices `color(blue)(P(1,3) and R(8,0)` of the diagonal PR

`Let " "(x_1, y_1) = P(1,3)`

`Let " "(x_2, y_2) = R(8,0)`

Using the Midpoint formula we can write

`[(1+8)/2, (3+0)/2]`

`[9/2, 3/2]`

`color(red)([4.5, 1.5]" "` Intermediate result.1

`color(green)(Step.2`

Consider the Vertices `color(blue)(Q(6,5) and S(3,-2)` of the diagonal QS

`Let " "(x_1, y_1) = Q(6,5)`

`Let " "(x_2, y_2) = S(3,-2)`

Using the Midpoint formula we can write

`[(6+3)/2, (5+(-2))/2]`

`[9/2, 3/2]`

`color(red)([4.5, 1.5]" "` Intermediate result.2

By observing the two Intermediate results 1 and 2, we understand that both the diagonals have the same Midpoint, and hence the given Quadrilateral with four vertices is a Parallelogram.

`color(green)(Step.3`

Please refer to the image of the graph constructed using GeoGebra given below:

MPPR `rArr` MidPoint of diagonal PR

MPQS `rArr` MidPoint of diagonal QS

`color(green)(Step.4`

Some interesting properties of a parallelogram to remember:

1. Opposite sides of a parallelogram have the same length and hence they are congruent.

2. Opposite angles of the parallelogram have the same size/measure.

3. Obviously, opposite sides of a parallelogram are also parallel.

4. The diagonals of a parallelogram bisect each other.

5. Each diagonal of a parallelogram separates it into two congruent triangles.

6. We observe that our parallelogram has all sides congruent, and hence our parallelogram is a rhombus.