Worksheet: Rectangles on 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 𝐴𝐡𝐢𝐷. Give your solution to two decimal places.

Work out the area of the rectangle 𝐴𝐡𝐢𝐷.

Q2:

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

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 𝐴𝐡𝐢𝐷, determine the values of π‘₯ and 𝑦.

  • Aπ‘₯=βˆ’1, 𝑦=βˆ’7
  • Bπ‘₯=βˆ’3, 𝑦=βˆ’7
  • Cπ‘₯=βˆ’3, 𝑦=7
  • Dπ‘₯=βˆ’1, 𝑦=9
  • Eπ‘₯=1, 𝑦=9

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 𝐴𝐢.

Work out the length of the diagonal 𝐡𝐷.

  • A2√2
  • B√2
  • C2√5
  • D6√2
  • E2√10

Using these lengths, is the parallelogram 𝐴𝐡𝐢𝐷 a rectangle?

  • Ano
  • Byes

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 triangle at 𝐡, find the coordinates of the point 𝐷 that make 𝐴𝐡𝐢𝐷 a rectangle.

  • Aο€Ό32,βˆ’2
  • B(βˆ’1,1)
  • C(9,1)
  • 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 𝐴𝐡𝐢𝐷 a rectangle.

  • A(7,βˆ’11)
  • B(8,βˆ’2)
  • C(6,βˆ’8)
  • D(8,βˆ’6)

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?

  • A1 by 2
  • B7 by 6
  • C7 by 2
  • D1 by 6
  • E6 by 2

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

Q11:

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

  • A|π‘Žβˆ’π‘‘|β‹…|π‘βˆ’π‘”|
  • B|π‘Žβˆ’π‘|β‹…|π‘βˆ’π‘‘|
  • C|π‘”βˆ’π‘’|β‹…|β„Žβˆ’π‘“|
  • D|π‘Žβˆ’π‘|β‹…|π‘‘βˆ’π‘“|
  • E|π‘”βˆ’π‘’|β‹…|β„Žβˆ’π‘’|

Q12:

Given that 𝐴𝐡𝐢𝑂 is a rectangle, find the length of 𝐴𝐢 and the area of the rectangle.

  • A5 length units, 169 area units
  • B17 length units, 144 area units
  • C√119 length units, 34 area units
  • D12 length units, 25 area units
  • E13 length units, 60 area units

Q13:

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 a parallelogram.
  • BIt is not a parallelogram.

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

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

Q14:

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 𝐴𝐢.

Work out the length of the diagonal 𝐡𝐷.

Using these lengths, is the parallelogram 𝐴𝐡𝐢𝐷 a rectangle?

  • Ayes
  • Bno

Q15:

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

Calculate the length of 𝐴𝐢. Give an exact answer.

  • A√10
  • B√250
  • C7
  • D13

Calculate the length of 𝐡𝐷. Give an exact answer.

  • A√250
  • B7
  • C13
  • D√10

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

  • AIt is a rectangle.
  • BWe cannot decide without more information.
  • CIt is not a rectangle.

Q16:

A parallelogram 𝐴𝐡𝐢𝐷 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 𝐴𝐡.

Work out the slope of 𝐡𝐢.

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

Is 𝐴𝐡𝐢𝐷 a rectangle?

  • Ano
  • Byes

Q17:

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.

  • ASlope 𝐴𝐡=112, slope 𝐡𝐢=43, slope 𝐢𝐷=2, slope 𝐴𝐷=12
  • BSlope 𝐴𝐡=12, slope 𝐡𝐢=βˆ’2, slope 𝐢𝐷=12, slope 𝐴𝐷=βˆ’2
  • CSlope 𝐴𝐡=βˆ’12, slope 𝐡𝐢=2, slope 𝐢𝐷=βˆ’12, slope 𝐴𝐷=2
  • DSlope 𝐴𝐡=2, slope 𝐡𝐢=βˆ’12, slope 𝐢𝐷=43, slope 𝐴𝐷=3
  • ESlope 𝐴𝐡=211, slope 𝐡𝐢=34, slope 𝐢𝐷=βˆ’2, slope 𝐴𝐷=βˆ’12

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

  • AYes
  • BNo

Q18:

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?

  • Arectangle
  • Brhombus
  • Cparallelogram
  • Dsquare
  • Etrapezoid

Q19:

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).

  • ASquare
  • BRectangle
  • CTriangle
  • DRhombus

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