Worksheet: Rectangles in the Coordinate Plane

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In this worksheet, we will practice using the distance, slope, and midpoint formulas to determine the coordinates, area, and perimeter of a rectangle in the coordinate plane.

Q1:

A rectangle has vertices at the points 𝐴 , 𝐵 , 𝐶 , and 𝐷 with coordinates ( 1 , 1 ) , ( 4 , 2 ) , ( 6 , − 4 ) , and ( 3 , − 5 ) respectively.

Work out the perimeter of the rectangle 𝐴 𝐵 𝐶 𝐷 . Give your solution to two decimal places.

Work out the area of the rectangle 𝐴 𝐵 𝐶 𝐷 .

Q2:

Rectangle 𝐴 𝐵 𝐶 𝐷 is graphed in the coordinate plane with its vertices at 𝐴 ( 0 , 0 ) , 𝐵 ( − 7 , 0 ) , 𝐶 ( − 7 , − 4 ) , and 𝐷 ( 0 , − 4 ) . Find its perimeter.

Q3:

Rectangle 𝐴 𝐵 𝐶 𝐷 is graphed in the coordinate plane with its vertices at 𝐴 ( 0 , 0 ) , 𝐵 ( 6 , 0 ) , 𝐶 ( 6 , 5 ) , and 𝐷 ( 0 , 5 ) . Find its area.

Q4:

Rectangle 𝐴 𝐵 𝐶 𝐷 is graphed in the coordinate plane with its vertices at 𝐴 ( − 3 , 4 ) , 𝐵 ( 5 , 4 ) , 𝐶 ( 5 , − 2 ) , and 𝐷 ( − 3 , − 2 ) . Find its area.

Q5:

Rectangle 𝐴 𝐵 𝐶 𝐷 is graphed in the coordinate plane with vertices 𝐴 ( − 5 , − 2 ) , 𝐵 ( 6 , − 2 ) , and 𝐶 ( 6 , 3 ) . Find the coordinates of point 𝐷 .

  • A ( − 3 , 6 )
  • B ( 5 , − 3 )
  • C ( 5 , 2 )
  • D ( − 5 , 3 )
  • E ( 2 , − 6 )

Q6:

Given that the points 𝐴 ( 3 , − 6 ) , 𝐵 ( 1 , 2 ) , 𝐶 ( − 3 , − 𝑥 ) , and 𝐷 ( 𝑥 , 𝑦 ) are the vertices of the rectangle 𝐴 𝐵 𝐶 𝐷 , determine the values of 𝑥 and 𝑦 .

  • A 𝑥 = − 3 , 𝑦 = 7
  • B 𝑥 = 1 , 𝑦 = 9
  • C 𝑥 = − 1 , 𝑦 = 9
  • D 𝑥 = − 1 , 𝑦 = − 7
  • E 𝑥 = − 3 , 𝑦 = − 7

Q7:

A parallelogram has vertices at the coordinates 𝐴 ( 1 , − 3 ) , 𝐵 ( 2 , − 1 ) , 𝐶 ( 5 , − 3 ) , and 𝐷 ( 4 , − 5 ) .

Work out the length of the diagonal 𝐴 𝐶 .

Work out the length of the diagonal 𝐵 𝐷 .

  • A 2 √ 5
  • B √ 2
  • C 2 √ 2
  • D 2 √ 1 0
  • E 6 √ 2

Using these lengths, is the parallelogram 𝐴 𝐵 𝐶 𝐷 a rectangle?

  • Ayes
  • Bno

Q8:

Given that the points 𝐴 ( − 5 , − 3 ) , 𝐵 ( − 4 , − 1 ) , and 𝐶 ( 0 , − 3 ) are the vertices of a right-angled triangle at 𝐵 , find the coordinates of the point 𝐷 that make 𝐴 𝐵 𝐶 𝐷 a rectangle.

  • A ( 9 , 1 )
  • B ( − 1 , 1 )
  • C 3 2 , − 2
  • D ( − 1 , − 5 )

Q9:

The coordinates of 𝐴 , 𝐵 , and 𝐶 are ( 8 , − 7 ) , ( 4 , − 7 ) , and ( 4 , − 6 ) respectively. Determine the coordinates of 𝐷 that would make 𝐴 𝐵 𝐶 𝐷 a rectangle.

  • A ( 7 , − 1 1 )
  • B ( 6 , − 8 )
  • C ( 8 , − 2 )
  • D ( 8 , − 6 )

Q10:

Nabil has a rectangle-shaped garden which he wants to fence. He sketched his garden on a coordinate plane with vertices at , (4,3), , and

What are the dimensions of his garden?

  • A6 by 2
  • B7 by 2
  • C1 by 2
  • D7 by 6
  • E1 by 6

If each unit of the grid represents 1 m, how many meters of fencing should he buy?

  • A 26 m
  • B 6 m
  • C 16 m
  • D 18 m
  • E 14 m

Q11:

Select the expression that is equal to the area of the rectangle.

  • A | 𝑔 − 𝑒 | ⋅ | ℎ − 𝑒 |
  • B | 𝑎 − 𝑑 | ⋅ | 𝑏 − 𝑔 |
  • C | 𝑎 − 𝑐 | ⋅ | 𝑏 − 𝑑 |
  • D | 𝑎 − 𝑐 | ⋅ | 𝑑 − 𝑓 |
  • E | 𝑔 − 𝑒 | ⋅ | ℎ − 𝑓 |

Q12:

Amira drew plans for her living room. She connected the following points, on grid paper, in order: ( 1 , 1 ) , ( 1 , 6 ) , ( 5 , 6 ) , ( 5 , 2 ) , ( 6 , 2 ) , ( 6 , 1 ) , and ( 1 , 1 ) . If the side length of each grid square represents 3 feet of the room, and Amira wants to carpet the room, how many square feet will she need to carpet?

Q13:

Given that 𝐴 𝐵 𝐶 𝑂 is a rectangle, find the length of 𝐴 𝐶 and the area of the rectangle.

  • A12 length units, 25 area units
  • B17 length units, 144 area units
  • C5 length units, 169 area units
  • D13 length units, 60 area units
  • E √ 1 1 9 length units, 34 area units

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