Worksheet: Diagonals of Parallelograms

In this worksheet, we will practice correlating between the diagonals of parallelograms and using these properties to find unknown measures.

Q1:

Given that 𝐶 𝑀 = 1 6 c m , determine the length of 𝐴 𝐶 .

Q2:

Find the value of 𝑧 in the following parallelogram.

Q3:

In parallelogram 𝐴 𝐵 𝐶 𝐷 , 𝐵 𝐶 = 8 9 , 𝑀 𝐵 = 4 6 , and 𝑀 𝐶 = 7 8 . What is the perimeter of 𝐴 𝑀 𝐷 ?

Q4:

What can you say about the diagonals of a parallelogram?

  • AThey are equal in length.
  • BThey are perpendicular.
  • CThey bisect each other.

Q5:

Is any quadrilateral whose diagonals bisect each other a parallelogram?

  • Ano
  • Byes

Q6:

Are the two diagonals of a parallelogram perpendicular?

  • Ayes
  • Bno

Q7:

If 𝑄 𝑅 𝑆 𝑇 is a parallelogram, find the value of 𝑧 .

Q8:

The given figure shows a parallelogram 𝐴 𝐵 𝐶 𝐷 .

Using what you know about alternate angles, determine which angle will have the same measure as 𝐴 𝐵 𝐷 .

  • A 𝐵 𝐷 𝐶
  • B 𝐶 𝐵 𝐴
  • C 𝐵 𝐷 𝐴
  • D 𝐵 𝐶 𝐷
  • E 𝐵 𝐴 𝐷

Which angle will be equal in measure to 𝐴 𝐷 𝐵 ?

  • A 𝐶 𝐵 𝐴
  • B 𝐵 𝐷 𝐶
  • C 𝐶 𝐵 𝐷
  • D 𝐵 𝐴 𝐷
  • E 𝐴 𝐷 𝐶

𝐵 𝐷 is a common side to both triangles. Using the information from the earlier parts of the question, can we prove that triangles 𝐴 𝐵 𝐷 and 𝐶 𝐷 𝐵 are congruent? If yes, by which congruence criteria?

  • ANo
  • BYes, by SAS
  • CYes, by ASA
  • DYes, by HL
  • EYes, by SSS

What will be true of 𝐴 𝐵 and 𝐶 𝐷 and of 𝐵 𝐶 and 𝐴 𝐷 ?

  • A 𝐴 𝐵 𝐶 𝐷 , 𝐴 𝐷 𝐵 𝐶
  • B 𝐴 𝐵 = 𝐶 𝐷 , 𝐴 𝐷 𝐵 𝐶
  • C 𝐴 𝐵 = 𝐶 𝐷 , 𝐴 𝐷 = 𝐵 𝐶
  • D 𝐴 𝐵 𝐶 𝐷 , 𝐴 𝐷 = 𝐵 𝐶

What will be true of angles 𝐵 𝐴 𝐷 and 𝐵 𝐶 𝐷 and of angles 𝐴 𝐵 𝐶 and 𝐴 𝐷 𝐶 ?

  • A 𝑚 𝐵 𝐴 𝐷 = 𝑚 𝐵 𝐶 𝐷 , 𝑚 𝐴 𝐵 𝐶 𝑚 𝐴 𝐷 𝐶
  • B 𝑚 𝐵 𝐴 𝐷 = 𝑚 𝐵 𝐶 𝐷 , 𝑚 𝐴 𝐵 𝐶 = 𝑚 𝐴 𝐷 𝐶
  • C 𝑚 𝐵 𝐴 𝐷 𝑚 𝐵 𝐶 𝐷 , 𝑚 𝐴 𝐵 𝐶 𝑚 𝐴 𝐷 𝐶
  • D 𝑚 𝐵 𝐴 𝐷 𝑚 𝐵 𝐶 𝐷 , 𝑚 𝐴 𝐵 𝐶 = 𝑚 𝐴 𝐷 𝐶

Q9:

Given that 𝐴 𝐵 𝐶 𝐷 is a parallelogram and 𝐶 𝑀 = 8 . 6 c m , find the perimeter of 𝐴 𝐵 𝐶 .

Q10:

In a parallelogram 𝐴 𝐵 𝐶 𝐷 , the coordinates of 𝐴 , 𝐵 , and 𝐶 are ( 9 , 0 ) , ( 1 1 , 0 ) , and ( 4 , 9 ) , respectively. Find the coordinates of the point at which the two diagonals intersect, and then determine the coordinates of point 𝐷 .

  • A ( 5 , 9 ) , ( 6 , 9 )
  • B 1 3 2 , 9 2 , ( 6 , 9 )
  • C 5 2 , 9 2 , ( 6 , 9 )
  • D ( 1 3 , 9 ) , ( 6 , 9 )

Q11:

𝐴 𝐵 𝐶 𝐷 is a parallelogram where the coordinates of 𝐴 are ( 7 , 7 ) and the coordinates of 𝐶 are ( 1 , 5 ) . Find the coordinates of the point of intersection of the two diagonals of 𝐴 𝐵 𝐶 𝐷 .

  • A ( 3 , 6 )
  • B ( 7 , 7 )
  • C ( 4 , 6 )
  • D ( 1 , 5 )

Q12:

In parallelogram 𝐴 𝐵 𝐶 𝐷 , the coordinates of 𝐴 are ( 1 , 3 ) and those of the intersection of the diagonals are ( 4 , 3 ) . What are the coordinates of point 𝐶 ?

  • A ( 9 , 7 )
  • B 1 2 , 2
  • C ( 7 , 9 )
  • D ( 3 , 6 )
  • E 5 2 , 0

Q13:

Find the value of 𝑧 in the following parallelogram.

Q14:

In parallelogram 𝐴 𝐵 𝐶 𝐷 , 𝐵 𝐶 = 4 7 , 𝑀 𝐵 = 4 3 , and 𝑀 𝐶 = 3 3 . What is the perimeter of 𝐴 𝑀 𝐷 ?

Q15:

If 𝑄 𝑅 𝑆 𝑇 is a parallelogram, find the value of 𝑧 .

Q16:

𝐴 𝐵 𝐶 𝐷 is a parallelogram where the coordinates of 𝐴 are ( 7 , 5 ) and the coordinates of 𝐶 are ( 1 , 3 ) . Find the coordinates of the point of intersection of the two diagonals of 𝐴 𝐵 𝐶 𝐷 .

  • A ( 4 , 4 )
  • B ( 3 , 4 )
  • C ( 7 , 5 )
  • D ( 1 , 3 )

Q17:

Given that 𝐴 𝐵 𝐶 𝐷 is a parallelogram and 𝐶 𝑀 = 5 . 8 c m , find the perimeter of 𝐴 𝐵 𝐶 .

Q18:

In a parallelogram 𝐴 𝐵 𝐶 𝐷 , the coordinates of 𝐴 , 𝐵 , and 𝐶 are ( 8 , 5 ) , ( 6 , 5 ) , and ( 1 3 , 8 ) , respectively. Find the coordinates of the point at which the two diagonals intersect, and then determine the coordinates of point 𝐷 .

  • A ( 5 , 1 3 ) , ( 1 , 8 )
  • B ( 2 1 , 3 ) , ( 1 , 8 )
  • C 2 1 2 , 3 2 , ( 1 , 8 )
  • D 5 2 , 1 3 2 , ( 1 , 8 )

Q19:

The perimeter of the rectangle is the same as the perimeter of the square.

What is the area of the square?

Q20:

In the figure below, given that 𝐴 𝐵 𝐶 𝐷 is a rectangle, 𝑂 𝐵 𝐶 𝐻 is a parallelogram, and 𝐴 𝑂 = 3 . 7 c m , find the length of 𝐻 𝐴 .

Q21:

If 𝐹 𝐽 = 6 𝑥 + 8 𝑦 , 𝐹 𝑀 = 3 𝑥 + 5 𝑦 , 𝐺 𝐻 = 4 2 , and 𝐺 𝑀 = 2 4 , what values of 𝑥 and 𝑦 make parellelogram 𝐹 𝐺 𝐻 𝐽 a rectangle?

  • A 𝑥 = 3 , 𝑦 = 3
  • B 𝑥 = 4 2 , 𝑦 = 2 4
  • C 𝑥 = 4 2 , 𝑦 = 4 8
  • D 𝑥 = 4 2 , 𝑦 = 3
  • E 𝑥 = 3 , 𝑦 = 2 4

Q22:

Given that 𝐴 𝐵 𝐶 𝐷 is a rectangle, where 𝐸 𝐶 = 6 𝑥 7 and 𝐴 𝐸 = 2 𝑥 + 5 , find 𝐷 𝐸 .

Q23:

In this rectangle, 𝑋 𝑍 = 7 𝑐 , 𝑍 𝑌 = 2 1 , and 𝑋 𝑌 = 2 8 . Find the value of 𝑐 .

Q24:

The lengths of two adjacent sides of a parallelogram are 8 𝑡 units and 8 𝑐 units. Write an expression for its perimeter.

  • A ( 1 6 𝑡 + 8 𝑐 ) units
  • B ( 8 𝑡 + 8 𝑐 ) units
  • C 2 ( 𝑡 + 𝑐 ) units
  • D 2 ( 8 𝑡 + 8 𝑐 ) units
  • E ( 8 𝑡 + 1 6 𝑐 ) units

Q25:

𝐴 𝐵 𝐶 𝐷 is a parallelogram. Given that the perimeter of 𝐴 𝐵 𝐶 𝐷 is 54, find the length of 𝐴 𝐵 .

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