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Worksheet: Proving That Parallelograms Are Rectangles

Q1:

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.

  • A13
  • B √ 1 0
  • C7
  • D √ 2 5 0

Calculate the length of 𝐡 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.

Q2:

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

Q3:

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.

Q4:

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?

  • Ayes
  • Bno