Question

Prove that quadrilateral Q(1, 2), U(2, 5), A(5, 7) and D(4, 4) is a parallelogram by using slopes.

Slope of QU =

Slope of AD =

Slope of UA =

Slope of QD =

Which sides are parallel (give the letters) A) sides QU and AD B) sides QU and UA C) sides UA and QD D) sides AD and QD

Answer 1

`m = 3` ---- Slopes of QU and AD

`m =(2)/(3)` ---- Slopes of UA and QD

Sides QU and AD have the same slope

Sides UA and QD have the same slope

Explanation:

Given

`Q = (1, 2)`

`U = (2, 5)`

`A = (5, 7)`

`D = (4, 4)`

Solving (a): Slope of QU

`Q = (x_1,y_1)= (1, 2)`

`U = (x_2,y_2)= (2, 5)`

Slope, m is calculated as:

`m = (y_2 - y_1)/(x_2 - x_1)`

`m =(5-2)/(2-1)`

`m =(3)/(1)`

`m = 3`

Slope of AD

`A =(x_1,y_1) = (5, 7)`

`D = (x_2,y_2) = (4, 4)`

Slope, m is calculated as:

`m = (y_2 - y_1)/(x_2 - x_1)`

`m =(4-7)/(4-5)`

`m =(-3)/(-1)`

`m = 3`

Slope of UA

`A =(x_1,y_1) = (5, 7)`

`U = (x_2,y_2)= (2, 5)`

Slope, m is calculated as:

`m = (y_2 - y_1)/(x_2 - x_1)`

`m =(5-7)/(2-5)`

`m =(-2)/(-3)`

`m =(2)/(3)`

Slope of QD

`Q = (x_1,y_1)= (1, 2)`

`D = (x_2,y_2) = (4, 4)`

Slope, m is calculated as:

`m = (y_2 - y_1)/(x_2 - x_1)`

`m =(4-2)/(4-1)`

`m =(2)/(3)`

Solving (b): Parallel Sides

Two sides are said to be parallel if they have the same slope.

The slope were calculated in (a) above and from there, we have the following observations

1. QU and AD have the same slope of 3

2. UA and QD have the same slope of 2/3