Worksheet: Parallelograms in the Coordinate Plane

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 ( βˆ’ 5 , 0 ) , 𝐿 ( βˆ’ 3 , βˆ’ 1 ) , 𝑀 ( βˆ’ 2 , 5 ) , and 𝑁 ( βˆ’ 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 𝐡 𝐢 𝐷 is a quadrilateral, 𝐴 = ( βˆ’ 2 , βˆ’ 1 7 ) , 𝐡 = ( βˆ’ 1 4 , 1 0 ) , 𝐢 = ( 1 , 7 ) , and 𝐷 = ( 1 3 , βˆ’ 2 0 ) , find the midpoint of 𝐴 𝐢 and 𝐡 𝐷 , then determine what type of figure 𝐴 𝐡 𝐢 𝐷 is.

  • A ( βˆ’ 1 , βˆ’ 5 ) , ( βˆ’ 1 , βˆ’ 5 ) , trapezium
  • B ο€Ό 5 2 , βˆ’ 8  , ο€Ό βˆ’ 1 7 , 2 3 2  , parallelogram
  • C ο€Ό βˆ’ 1 2 , βˆ’ 1 0  , ο€Ό βˆ’ 1 2 , βˆ’ 1 0  , trapezium
  • D ο€Ό βˆ’ 1 2 , βˆ’ 5  , ο€Ό βˆ’ 1 2 , βˆ’ 5  , parallelogram

Q3:

Where must the coordinates of point 𝐢 be so that 𝐴 𝐡 𝐢 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 𝐡 𝐢 𝐷 is a parallelogram, what can be said of the slope of line βƒ–     βƒ— 𝐴 𝐡 ?

  • Athe slope of line βƒ–     βƒ— 𝐴 𝐡 = the slope of line βƒ–     βƒ— 𝐴 𝐢
  • Bthe slope of line βƒ–     βƒ— 𝐴 𝐡 = the slope of line βƒ–     βƒ— 𝐡 𝐢
  • Cthe slope of line βƒ–     βƒ— 𝐴 𝐡 = the slope of line βƒ–      βƒ— 𝐴 𝐷
  • Dthe slope of line βƒ–     βƒ— 𝐴 𝐡 = the slope of line βƒ–     βƒ— 𝐢 𝐷

Q5:

𝐴 𝐡 𝐢 𝐷 is a parallelogram. The coordinates of the points 𝐴 , 𝐡 , and 𝐢 are ( 0 , βˆ’ 2 ) , ( 4 , 7 ) , and ( 6 , 3 ) respectively. Find the coordinates of 𝐷 .

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

Q6:

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

Q7:

Given that 𝐿 = ( βˆ’ 5 , βˆ’ 6 , 0 ) , 𝑀 = ( βˆ’ 2 , βˆ’ 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 , 𝐡 , 𝐢 , and 𝐷 with coordinates ( βˆ’ 1 , 1 ) , ( 1 , 3 ) , ( 3 , βˆ’ 1 ) , and ( 1 , βˆ’ 3 ) respectively.

Work out the perimeter of the parallelogram 𝐴 𝐡 𝐢 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 𝐡 𝐢 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 ( βˆ’ 5 , 1 ) , 𝐿 ( 1 , 0 ) , 𝑀 ( 3 , βˆ’ 2 ) , and 𝑁 ( βˆ’ 3 , βˆ’ 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 𝐢 be so that 𝐴 𝐡 𝐢 𝐷 is a parallelogram? In that case, what is the area of the parallelogram?

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

Q13:

Where must the coordinates of point 𝐢 be so that 𝐴 𝐡 𝐢 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 𝐡 𝐢 𝐷 is a quadrilateral, 𝐴 = ( βˆ’ 8 , 1 ) , 𝐡 = ( 8 , 4 ) , 𝐢 = ( βˆ’ 2 , 8 ) , and 𝐷 = ( βˆ’ 1 8 , 5 ) , find the midpoint of 𝐴 𝐢 and 𝐡 𝐷 , then determine what type of figure 𝐴 𝐡 𝐢 𝐷 is.

  • A ο€Ό βˆ’ 1 0 , 9 2  , ο€Ό βˆ’ 1 0 , 9 2  , trapezium
  • B ο€Ό 0 , βˆ’ 1 2  , ο€Ό 1 3 2 , βˆ’ 7  , parallelogram
  • C ( βˆ’ 5 , 9 ) , ( βˆ’ 5 , 9 ) , trapezium
  • D ο€Ό βˆ’ 5 , 9 2  , ο€Ό βˆ’ 5 , 9 2  , parallelogram

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