Diagonals of a parallelogram
Do the diagonals cut each other in half?
Cut in half?
In a rectangle the two diagonals cross at their middle points. A leaning parallelogram has two diagonals too: one long, one short.
Where do the two diagonals of a parallelogram cross?
What this lesson covers
The idea
The diagonals of a parallelogram bisect each other, though they need not be equal.
Cut in half?
In a rectangle the two diagonals cross at their middle points. A leaning parallelogram has two diagonals too: one long, one short.
Where do the two diagonals of a parallelogram cross?
- Each one is cut in half by the other
- Only the longer one is cut in half
- Neither one is cut in half
Find the middles
Both diagonals are drawn, with a pin on each. Slide each pin to the middle of its diagonal: the two parts must be equal. Where do the pins end up? Do this for each parallelogram.
One crossing point
The diagonals of a parallelogram bisect each other, though they need not be equal.
In every parallelogram the two middle pins met at one point O: each diagonal is cut into two equal halves by the other.
Why? In ∆AOB and ∆COD: AB = CD (opposite sides), and ∠OAB = ∠OCD, ∠OBA = ∠ODC (alternate angles, AB ∥ DC). So ∆AOB ≅ ∆COD by ASA, and OA = OC, OB = OD.
But the diagonals were not equal: only in the rectangle were they the same length.
Notes
The diagonals of a parallelogram bisect each other, though they need not be equal.
Check yourself
The diagonals of parallelogram ABCD meet at O. AC = 12 cm. How long is OA?
Answer: 6 cm
The diagonals bisect each other, so O is the midpoint of AC. OA = 12 ÷ 2 = 6 cm.
The diagonals of parallelogram ABCD meet at O. OA = 2x + 1 and OC = 11. What is x?
Answer: 5
The diagonals bisect each other, so OA = OC. Then 2x + 1 = 11, so 2x = 10 and x = 5.
In parallelogram PQRS the diagonals are PR = 10 cm and QS = 6 cm, meeting at O. Which is true?
Which statement is not always true for a parallelogram?
- PO = 5 cm and QO = 3 cm — correct. Yes! Each diagonal is cut in half by the other: PO = 10 ÷ 2 = 5 cm and QO = 6 ÷ 2 = 3 cm.
- PO = 5 cm and QO = 5 cm. The diagonals bisect each other, but they are different lengths, so the halves differ too: QO is half of 6 cm.
- This cannot happen: the diagonals of a parallelogram must be equal. The diagonals of a parallelogram need not be equal. Only in a rectangle are they always equal.
- The diagonals bisect each other. This is always true for a parallelogram.
- The diagonals are equal — correct. Yes! The diagonals of a parallelogram need not be equal. In a leaning one, one diagonal is longer.
- The opposite sides are equal. This is always true for a parallelogram.