On a piece of graph paper, plot the following points: A (3, 1), B (1, 5), C (9, 9), and D (11, 5). These coordinates will be the

Question

2 years ago

11

vertices of a quadrilateral. How would you use the distance formula and the slope formula to prove that this figure is actually a rectangle?

1 answer:

yarga [219] · Answer 1 · 2023-12-02T17:28:01+03:00

The given coordinates are actually a rectangle

<h3>How to determine the quadrilateral type?</h3>

The coordinates are given as:

A (3, 1), B (1, 5), C (9, 9), and D (11, 5).

Calculate the distance between the coordinates using:

$d = \sqrt{(x_2 -x_1)^2 +(y_2 -y_1)^2$

So, we have:

$AB = \sqrt{(3 -1)^2 +(1-5)^2} =\sqrt {20$

$BC = \sqrt{(1 -9)^2 +(5-9)^2} =\sqrt {80$

$CD = \sqrt{(9 -11)^2 +(9-5)^2} =\sqrt {20$

$DA = \sqrt{(11 -3)^2 +(5-1)^2} =\sqrt {80$

The above shows that the opposite sides are congruent

Next, we calculate the slopes using:

m = (y2- y1)/(x2- x1)

So, we have:

AB = (1- 5)/(3-1) = -2

BC = (5- 9)/(1-9) = 1/2

CD = (9- 5)/(9-11) = -2

DA = (5- 1)/(11-3) = 1/2

The slopes of adjacent sides are opposite reciprocals.

This means that the sides are perpendicular

Hence, the given coordinates are actually a rectangle