Question

How do you verify that parallelogram ABCD with vertices A(-5. -1), B(-9, 6), C(-1. 5). and D(3,-2) is a rhombus by showing that it is a parallelogram with perpendicular diagonal?

Answer 1

Hence it’s a `color(blue)(RHOMBUS`

Explanation:

`A(-5,-1), B(-9,6), C(-1,5), D(3,-2)`

Properties of a rhombus :
a) opposite sides are parallel. b) All the four sides are equal.
c) Diagonals bisect each other at right angles.

If AB // CD, BC // AD, it’s a parallelogram.

If slope of AB = CD, BC = AD then it’s a parallelogram.

`m_(AB) = (6+1) / (-9+5) = -7/4`

`m_(CD) = (-2-5) / (3+1) = -7/4`

`m_(BC) =( 5-6) / (-1+9) = -1/8`

`m_(AD) = (-2+1) / (3+5) = -1/7`

Hence AB // CD, BC // AD and ABCD is a parallelogram

Slope of AC `m_(AC) = (5+1) / (-1+5) = 3/2`

Slope of BD `m_(BD) = (6+2) / (-9-3) = -2/3`

`m_(AC) = -1/m_(BD)`

Hence diagonals intersect each other at right angles.

It can be a square or a rhombus or a kite.

It’s not a square since AB not perpendicular AD.

It can be a rhombus or a kite.

If the intersection point of the diagonals is their midpoint, then it’s a rhombus, else it’s a kite.

Let’s find the mid point (E) of AC & BD to prove it’s a rhombus.

`E = (A(-5,-1) + C(-1,5)) / 2 = (-3, 2)`

Also `E = (B(-9,6) + D(3,-2))/2 = (-3,2)`

Hence it’s a `color(blue)(RHOMBUS`