Question

The vertices of parallelogram ABCD are located at points A(-2,-1), B(6,1), C(10,7), and D(2,5). Which of the following statements are true?

Select all that apply.

Answer 1

The statements which are true are:

(6, 6) is the midpoint of CD

(4, 3) is the intersection point of diagonals of parallelogram

Solution:

The mid point (x,y) =
`( (x_1+x_2)/(2),(y_1+y_2)/(2))`

Midpoint of AB

A(-2, -1) and B(6, 1)

`\text{ midpoint of AB } = ((-2+6)/(2) , (-1+1)/(2))\\\\\text{ midpoint of AB } = (2, 0)`

Thus statement 1 is wrong

Midpoint of BC

B(6, 1) and C(10, 7)

`\text{ midpoint of BC } = ((6+10)/(2) , (1+7)/(2))\\\\\text{ midpoint of BC } = (8, 4)`

Thus statement 2 is wrong

Mid point of CD

Here ,

`x_1` = 10

`x_2`= 2

`y_1`= 7

`y_2`=5

now substituting these values,

mid point of CD =
`((10+2)/(2),(7+5)/(2))`

mid point of CD =
`((12)/(2),(12)/(2))`

mid point of CD =
`(6, 6)`

Therefore (6, 6) is the midpoint of CD

Statement 3 is correct

Midpoint of AD

A = (-2, -1) and D = (2, 5)

`\text{ mid point of AD } = ((-2+2)/(2) , (-1+5)/(2))\\\\\text{ mid point of AD } = (0, 2)`

Thus statement 4 is wrong

Intersection point of diagonals of parallelogram

Let AC and BD be the diagonals of parallelogram

The diagonals of a parallelogram bisect each other, therefore, the point of intersection is the midpoint of either.

Midpoint of AC:

A = (-2, -1) and C(10, 7)

`\text{ Midpoint of AC } = ((-2+10)/(2) , (-1+7)/(2))\\\\\text{ Midpoint of AC } = (4,3)`

Thus statement 5 is correct