# 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)?

Feb 14, 2018

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

$L e t \text{ } \left({x}_{1} , {y}_{1}\right) = P \left(1 , 3\right)$

$L e t \text{ } \left({x}_{2} , {y}_{2}\right) = R \left(8 , 0\right)$

Using the Midpoint formula we can write

$\left[\frac{1 + 8}{2} , \frac{3 + 0}{2}\right]$

$\left[\frac{9}{2} , \frac{3}{2}\right]$

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

$L e t \text{ } \left({x}_{1} , {y}_{1}\right) = Q \left(6 , 5\right)$

$L e t \text{ } \left({x}_{2} , {y}_{2}\right) = S \left(3 , - 2\right)$

Using the Midpoint formula we can write

$\left[\frac{6 + 3}{2} , \frac{5 + \left(- 2\right)}{2}\right]$

$\left[\frac{9}{2} , \frac{3}{2}\right]$

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 $\Rightarrow$ MidPoint of diagonal PR

MPQS $\Rightarrow$ 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.