Question

Ava wants to draw a parallelogram on the coordinate plane. She

plots these 3 points.

Part A
Find and label the coordinates of the fourth vertex, K, of the
parallelogram. Draw the parallelogram.

Part B
What is the length of side JK? How do you know?​

Answer 1

`k =(2,1)`

`JK = 2`

Explanation:

Given

`J = (0,1)` ----
`(x_1,y_1)`

`H = (1,-2)` ---
`(x_2,y_2)`

`I = (3,-2)` ---
`(x_3,y_3)`

See attachment for grid

Solving (a): The coordinates of K

The parallelogram has the following diagonals: IJ and HK

Diagonals bisect one another. So:

Midpoint of IJ = Midpoint of HK

This gives:

`(1)/(2)(I + J) = (1)/(2)(H+K)`

`(1)/(2)(x_3+x_1,y_3+y_1) = (1)/(2)(x_2+x,y_2+y)`

`(1)/(2)(3+0,-2+1) = (1)/(2)(1+x,-2+y)`

`(1)/(2)(3,-1) = (1)/(2)(1+x,-2+y)`

Multiply through by 2

`(3,-1) = (1+x,-2+y)`

By comparison:

`1 + x = 3`

`-2 + y = -1`

Solve for x and y

`x = 3 - 1 =2`

`y = -1 +2 = 1`

So, the coordinates of k is:

`k =(2,1)`

The length of JK is calculated using distance (d) formula

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

`J = (0,1)` ----
`(x_1,y_1)`

`k =(2,1)` ----
`(x_2,y_2)`

So:

`d = \sqrt{(0 - 2)^2 + (1 - 1)^2`

`d = \sqrt{(- 2)^2 + (0)^2`

`d = \sqrt{4 + 0`

`d = \sqrt{4`

`d = 2`

Hence:

`JK = 2`