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)?

1 Answer
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

#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:

enter image source here

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.