Worksheet: Parallelograms in the Coordinate Plane

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In this worksheet, we will practice using the distance, slope, and midpoint formulas to determine whether a quadrilateral in the coordinate plane is a parallelogram.

Q1:

The points 𝐾 ( − 5 , 0 ) , 𝐿 ( − 3 , − 1 ) , 𝑀 ( − 2 , 5 ) , and 𝑁 ( − 4 , 6 ) are the vertices of quadrilateral 𝐾 𝐿 𝑀 𝑁 . Using the slope formula, is the quadrilateral a parallelogram?

  • Ayes
  • Bno

Q2:

If 𝐴 𝐵 𝐶 𝐷 is a quadrilateral, 𝐴 = ( − 2 , − 1 7 ) , 𝐵 = ( − 1 4 , 1 0 ) , 𝐶 = ( 1 , 7 ) , and 𝐷 = ( 1 3 , − 2 0 ) , find the midpoint of 𝐴 𝐶 and 𝐵 𝐷 , then determine what type of figure 𝐴 𝐵 𝐶 𝐷 is.

  • A ( − 1 , − 5 ) , ( − 1 , − 5 ) , trapezium
  • B 5 2 , − 8 , − 1 7 , 2 3 2 , parallelogram
  • C − 1 2 , − 1 0 , − 1 2 , − 1 0 , trapezium
  • D − 1 2 , − 5 , − 1 2 , − 5 , parallelogram

Q3:

Where must the coordinates of point 𝐶 be so that 𝐴 𝐵 𝐶 𝐷 is a parallelogram? In that case, what is the area of the parallelogram?

  • A ( 5 , 6 ) , area = 24
  • B ( 6 , 5 ) , area = 24
  • C ( 5 , 6 ) , area = 35
  • D ( 6 , 5 ) , area = 35

Q4:

If 𝐴 𝐵 𝐶 𝐷 is a parallelogram, what can be said of the slope of line ⃖ ⃗ 𝐴 𝐵 ?

  • Athe slope of line ⃖ ⃗ 𝐴 𝐵 = the slope of line ⃖ ⃗ 𝐴 𝐶
  • Bthe slope of line ⃖ ⃗ 𝐴 𝐵 = the slope of line ⃖ ⃗ 𝐵 𝐶
  • Cthe slope of line ⃖ ⃗ 𝐴 𝐵 = the slope of line ⃖ ⃗ 𝐴 𝐷
  • Dthe slope of line ⃖ ⃗ 𝐴 𝐵 = the slope of line ⃖ ⃗ 𝐶 𝐷

Q5:

𝐴 𝐵 𝐶 𝐷 is a parallelogram. The coordinates of the points 𝐴 , 𝐵 , and 𝐶 are ( 0 , − 2 ) , ( 4 , 7 ) , and ( 6 , 3 ) respectively. Find the coordinates of 𝐷 .

  • A ( 2 , 8 )
  • B ( 1 0 , − 6 )
  • C ( 1 0 , 8 )
  • D ( 2 , − 6 )

Q6:

Suppose that ⃑ 𝐴 = ( − 3 , − 9 , − 9 ) and ⃑ 𝐵 = ( − 8 , − 7 , 5 ) fix two sides of a parallelogram. What is the area of this parallelogram, to the nearest hundredth?

Q7:

Given that 𝐿 = ( − 5 , − 6 , 0 ) , 𝑀 = ( − 2 , − 7 , 8 ) , and 𝑁 = ( 2 , 6 , 4 ) , determine the area of the parallelogram 𝐿 𝑀 𝑁 𝐸 to the nearest hundredth.

Q8:

Determine, in square units, the area of the shown parallelogram.

Q9:

A parallelogram has vertices at the points 𝐴 , 𝐵 , 𝐶 , and 𝐷 with coordinates ( − 1 , 1 ) , ( 1 , 3 ) , ( 3 , − 1 ) , and ( 1 , − 3 ) respectively.

Work out the perimeter of the parallelogram 𝐴 𝐵 𝐶 𝐷 . Give your solution to one decimal place.

By drawing a rectangle through the vertices of the parallelogram, or otherwise. Work out the area of the parallelogram 𝐴 𝐵 𝐶 𝐷 .

Q10:

Calculate, to two decimal places, the area of the parallelogram 𝑃 𝑄 𝑅 𝑆 , where the coordinates of its vertices are at 𝑃 ( 2 , 1 , 3 ) , 𝑄 ( 1 , 4 , 5 ) , 𝑅 ( 2 , 5 , 3 ) , and 𝑆 ( 3 , 2 , 1 ) .

Q11:

The points 𝐾 ( − 5 , 1 ) , 𝐿 ( 1 , 0 ) , 𝑀 ( 3 , − 2 ) , and 𝑁 ( − 3 , − 1 ) are the vertices of quadrilateral 𝐾 𝐿 𝑀 𝑁 . Using the slope formula, is the quadrilateral a parallelogram?

  • Ayes
  • Bno

Q12:

Where must the coordinates of point 𝐶 be so that 𝐴 𝐵 𝐶 𝐷 is a parallelogram? In that case, what is the area of the parallelogram?

  • A ( 4 , − 1 ) , area = 24
  • B ( − 1 , 4 ) , area = 24
  • C ( 4 , − 1 ) , area = 36
  • D ( − 1 , 4 ) , area = 36

Q13:

Where must the coordinates of point 𝐶 be so that 𝐴 𝐵 𝐶 𝐷 is a parallelogram? In that case, what is the area of the parallelogram?

  • A ( 3 , 6 ) , area = 28
  • B ( 6 , 3 ) , area = 28
  • C ( 3 , 6 ) , area = 49
  • D ( 6 , 3 ) , area = 49

Q14:

If 𝐴 𝐵 𝐶 𝐷 is a quadrilateral, 𝐴 = ( − 8 , 1 ) , 𝐵 = ( 8 , 4 ) , 𝐶 = ( − 2 , 8 ) , and 𝐷 = ( − 1 8 , 5 ) , find the midpoint of 𝐴 𝐶 and 𝐵 𝐷 , then determine what type of figure 𝐴 𝐵 𝐶 𝐷 is.

  • A − 1 0 , 9 2 , − 1 0 , 9 2 , trapezium
  • B 0 , − 1 2 , 1 3 2 , − 7 , parallelogram
  • C ( − 5 , 9 ) , ( − 5 , 9 ) , trapezium
  • D − 5 , 9 2 , − 5 , 9 2 , parallelogram

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