Worksheet: Proving That Parallelograms Are Rectangles

In this worksheet, we will practice determining if a parallelogram is a rectangle in terms of its sides, angles, and/or diagonals.

Q1:

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 a rectangle.
  • BIt is not a rectangle.

Q2:

A parallelogram 𝐴 𝐵 𝐶 𝐷 has vertices 𝐴 ( 5 , 5 ) , 𝐵 ( 9 , 3 ) , 𝐶 ( 8 , 4 ) , and 𝐷 ( 6 , 2 ) .

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

  • A13
  • B 1 0
  • C7
  • D 2 5 0

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

  • A7
  • B13
  • C 1 0
  • D 2 5 0

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

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

Q3:

A parallelogram has vertices at the coordinates 𝐴 ( 4 , 1 ) , 𝐵 ( 0 , 3 ) , 𝐶 ( 1 , 5 ) , and 𝐷 ( 5 , 3 ) .

Work out the length of the diagonal 𝐴 𝐶 .

Work out the length of the diagonal 𝐵 𝐷 .

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

  • Ano
  • Byes

Q4:

A parallelogram 𝐴 𝐵 𝐶 𝐷 has vertices at the coordinates 𝐴 ( 1 , 2 ) , 𝐵 ( 0 , 4 ) , 𝐶 ( 3 , 1 ) , and 𝐷 ( 2 , 1 ) .

Work out the slope of 𝐴 𝐵 .

Work out the slope of 𝐵 𝐶 .

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

Is 𝐴 𝐵 𝐶 𝐷 a rectangle?

  • Ayes
  • Bno

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