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Worksheet: Rectangles in the Coordinate Plane

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 ( 6 , 0 ) , 𝐢 ( 6 , 5 ) , and 𝐷 ( 0 , 5 ) . Find its area.

Q3:

Given that the points , , and are the vertices of a right triangle at , find the coordinates of the point that make a rectangle.

  • A
  • B
  • C
  • D

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:

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

Q6:

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

Q7:

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

Q8:

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?

  • 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

Q9:

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

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

Q10:

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

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

  • Ayes
  • Bno

Q11:

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?

Q12:

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.

Q13:

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