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 , 𝐡 , 𝐢 , and 𝐷 with coordinates ( 1 , 1 ) , ( 4 , 2 ) , ( 6 , βˆ’ 4 ) , and ( 3 , βˆ’ 5 ) respectively.

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

Work out the area of the rectangle 𝐴 𝐡 𝐢 𝐷 .

Q2:

Rectangle 𝐴 𝐡 𝐢 𝐷 is graphed in the coordinate plane with its vertices at 𝐴 ( 0 , 0 ) , 𝐡 ( βˆ’ 7 , 0 ) , 𝐢 ( βˆ’ 7 , βˆ’ 4 ) , and 𝐷 ( 0 , βˆ’ 4 ) . Find its perimeter.

Q3:

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

Q4:

Rectangle 𝐴 𝐡 𝐢 𝐷 is graphed in the coordinate plane with its vertices at 𝐴 ( βˆ’ 3 , 4 ) , 𝐡 ( 5 , 4 ) , 𝐢 ( 5 , βˆ’ 2 ) , and 𝐷 ( βˆ’ 3 , βˆ’ 2 ) . Find its area.

Q5:

Rectangle 𝐴 𝐡 𝐢 𝐷 is graphed in the coordinate plane with vertices 𝐴 ( βˆ’ 5 , βˆ’ 2 ) , 𝐡 ( 6 , βˆ’ 2 ) , and 𝐢 ( 6 , 3 ) . Find the coordinates of point 𝐷 .

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

Q6:

Given that the points 𝐴 ( 3 , βˆ’ 6 ) , 𝐡 ( 1 , 2 ) , 𝐢 ( βˆ’ 3 , βˆ’ π‘₯ ) , and 𝐷 ( π‘₯ , 𝑦 ) are the vertices of the rectangle 𝐴 𝐡 𝐢 𝐷 , determine the values of π‘₯ and 𝑦 .

  • A π‘₯ = βˆ’ 1 , 𝑦 = 9
  • B π‘₯ = βˆ’ 1 , 𝑦 = βˆ’ 7
  • C π‘₯ = 1 , 𝑦 = 9
  • D π‘₯ = βˆ’ 3 , 𝑦 = 7
  • E π‘₯ = βˆ’ 3 , 𝑦 = βˆ’ 7

Q7:

A parallelogram has vertices at the coordinates 𝐴 ( 1 , βˆ’ 3 ) , 𝐡 ( 2 , βˆ’ 1 ) , 𝐢 ( 5 , βˆ’ 3 ) , and 𝐷 ( 4 , βˆ’ 5 ) .

Work out the length of the diagonal 𝐴 𝐢 .

Work out the length of the diagonal 𝐡 𝐷 .

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

Using these lengths, is the parallelogram 𝐴 𝐡 𝐢 𝐷 a rectangle?

  • Ayes
  • Bno

Q8:

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

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

Q9:

The coordinates of 𝐴 , 𝐡 , and 𝐢 are ( 8 , βˆ’ 7 ) , ( 4 , βˆ’ 7 ) , and ( 4 , βˆ’ 6 ) respectively. Determine the coordinates of 𝐷 that would make 𝐴 𝐡 𝐢 𝐷 a rectangle.

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

Q10:

William has a rectangle-shaped garden which he wants to fence. He sketched his garden on a coordinate plane with vertices at ( βˆ’ 2 , 3 ) , (4,3), ( 4 , βˆ’ 4 ) , and ( βˆ’ 2 , βˆ’ 4 )

What are the dimensions of his garden?

  • A7 by 6
  • B6 by 2
  • C1 by 6
  • D1 by 2
  • E7 by 2

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

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

Q11:

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

  • A | π‘Ž βˆ’ 𝑐 | β‹… | 𝑏 βˆ’ 𝑑 |
  • B | π‘Ž βˆ’ 𝑐 | β‹… | 𝑑 βˆ’ 𝑓 |
  • C | 𝑔 βˆ’ 𝑒 | β‹… | β„Ž βˆ’ 𝑓 |
  • D | 𝑔 βˆ’ 𝑒 | β‹… | β„Ž βˆ’ 𝑒 |
  • E | π‘Ž βˆ’ 𝑑 | β‹… | 𝑏 βˆ’ 𝑔 |

Q12:

Sophia 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 Sophia wants to carpet the room, how many square feet will she need to carpet?

Q13:

Given that 𝐴 𝐡 𝐢 𝑂 is a rectangle, find the length of 𝐴 𝐢 and the area of the rectangle.

  • A √ 1 1 9 length units, 34 area units
  • B5 length units, 169 area units
  • C12 length units, 25 area units
  • D13 length units, 60 area units
  • E17 length units, 144 area units

Q14:

A quadrilateral has vertices at ( 0 , 3 ) , ( 1 , 5 ) , ( 5 , 3 ) , and ( 4 , 1 ) .

Decide if the quadrilateral is a parallelogram by calculating the length of each side.

  • AIt is not a parallelogram.
  • BIt is a parallelogram.

Decide if the quadrilateral is a rectangle by calculating the length of its diagonals.

  • AIt is a rectangle.
  • BIt is not a rectangle.

Q15:

A parallelogram has vertices at the coordinates 𝐴 ( βˆ’ 4 , βˆ’ 1 ) , 𝐡 ( 0 , βˆ’ 3 ) , 𝐢 ( βˆ’ 1 , βˆ’ 5 ) , and 𝐷 ( βˆ’ 5 , βˆ’ 3 ) .

Work out the length of the diagonal 𝐴 𝐢 .

Work out the length of the diagonal 𝐡 𝐷 .

Using these lengths, is the parallelogram 𝐴 𝐡 𝐢 𝐷 a rectangle?

  • Ano
  • Byes

Q16:

A parallelogram 𝐴 𝐡 𝐢 𝐷 has vertices 𝐴 ( βˆ’ 5 , 5 ) , 𝐡 ( 9 , 3 ) , 𝐢 ( 8 , βˆ’ 4 ) , and 𝐷 ( βˆ’ 6 , βˆ’ 2 ) .

Calculate the length of 𝐴 𝐢 . Give an exact answer.

  • A √ 1 0
  • B13
  • C7
  • D √ 2 5 0

Calculate the length of 𝐡 𝐷 . Give an exact answer.

  • A13
  • B √ 1 0
  • C √ 2 5 0
  • D7

Hence, state whether or not the parallelogram is a rectangle.

  • AWe cannot decide without more information.
  • BIt is not a rectangle.
  • CIt is a rectangle.

Q17:

A parallelogram 𝐴 𝐡 𝐢 𝐷 has vertices at the coordinates 𝐴 ( βˆ’ 1 , 2 ) , 𝐡 ( 0 , 4 ) , 𝐢 ( 3 , 1 ) , and 𝐷 ( 2 , βˆ’ 1 ) .

Work out the slope of 𝐴 𝐡 .

Work out the slope of 𝐡 𝐢 .

Work out the product of the slopes from parts (a) and (b).

Is 𝐴 𝐡 𝐢 𝐷 a rectangle?

  • Ano
  • Byes

Q18:

A quadrilateral has vertices at the points 𝐴 ( 0 , 5 ) , 𝐡 ( 2 , 6 ) , 𝐢 ( 4 , 2 ) , and 𝐷 ( 2 , 1 ) .

Work out the slope of the four edges of the quadrilateral.

  • A Slope 𝐴 𝐡 = 2 , slope 𝐡 𝐢 = βˆ’ 1 2 , slope 𝐢 𝐷 = 4 3 , slope 𝐴 𝐷 = 3
  • B Slope 𝐴 𝐡 = 2 1 1 , slope 𝐡 𝐢 = 3 4 , slope 𝐢 𝐷 = βˆ’ 2 , slope 𝐴 𝐷 = βˆ’ 1 2
  • C Slope 𝐴 𝐡 = 1 2 , slope 𝐡 𝐢 = βˆ’ 2 , slope 𝐢 𝐷 = 1 2 , slope 𝐴 𝐷 = βˆ’ 2
  • D Slope 𝐴 𝐡 = 1 1 2 , slope 𝐡 𝐢 = 4 3 , slope 𝐢 𝐷 = 2 , slope 𝐴 𝐷 = 1 2
  • E Slope 𝐴 𝐡 = βˆ’ 1 2 , slope 𝐡 𝐢 = 2 , slope 𝐢 𝐷 = βˆ’ 1 2 , slope 𝐴 𝐷 = 2

By finding the product of the slope, can we know if the quadrilateral is a rectangle?

  • ANo
  • BYes

Q19:

A quadrilateral has vertices at the points ( 2 , 1 ) , ( 3 , 3 ) , ( 5 , 2 ) , and ( 4 , 0 ) . By determining the lengths of the quadrilateral’s sides, and considering the gradients of the intersecting lines, what is the name of the quadrilateral?

  • Atrapezium
  • B rectangle
  • C parallelogram
  • D square
  • E rhombus

Q20:

Name the polygon that can be graphed in the coordinate plane with vertices at ( βˆ’ 2 , 3 ) , ( 3 , 3 ) , ( 3 , βˆ’ 1 ) , and ( βˆ’ 2 , βˆ’ 1 ) .

  • A Rhombus
  • B Rectangle
  • C Square
  • D Triangle

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