Where the strips cross
Find the middle of each diagonal.
Where do we join them?
The carpenter knows that both strips must be the same length. Now she must join them. Where should the two strips cross?
Where do the two diagonals of a rectangle cross?
What this lesson covers
The idea
The diagonals of a rectangle always intersect at their midpoints, that is, they bisect each other; bisecting a quantity means dividing it into two equal parts.
Where do we join them?
The carpenter knows that both strips must be the same length. Now she must join them. Where should the two strips cross?
Where do the two diagonals of a rectangle cross?
- Anywhere along the strips
- At the middle of both strips
- At the middle of only one strip
Find the middle
Each diagonal is a strip. Drag its pin until the two parts are equal. Then look at where the strips cross.
The middles meet
The diagonals of a rectangle always intersect at their midpoints, that is, they bisect each other. Bisecting a quantity means dividing it into two equal parts.
Why? In ∆AOB and ∆COD: ∠ABD = ∠CDB (both are 90° − ∠DBC) ∠AOB = ∠COD (vertically opposite) AB = CD (opposite sides) So they are congruent by AAS: OA = OC and OB = OD.
Notes
The diagonals of a rectangle always intersect at their midpoints: they bisect each other.
Check yourself
The diagonals of a rectangle cross at O. The diagonal AC is 14 cm. How long is OA?
Answer: 7 cm
O is the midpoint of AC, so OA is half of AC: 14 ÷ 2 = 7 cm.
Which sentence says that the diagonals of a rectangle bisect each other?
In rectangle ABCD the diagonals meet at O. Which is true?
In rectangle PQRS the diagonals meet at O and OP = 6.5 cm. How long is the diagonal QS?
Answer: 13 cm
O is the midpoint of PR, so PR = 2 × 6.5 = 13 cm. The diagonals are equal, so QS = 13 cm.
- They are equal in length. That is another property. Bisecting means dividing into two equal parts.
- Each diagonal is cut into two equal parts by the other — correct. Yes! The crossing point is the midpoint of each diagonal.
- They cross at 90°. The diagonals of a rectangle cross at 90° only when it is a square. Bisecting is about equal parts.
- OA = OC only. OA = OC is true, but there is more. The diagonals are also equal, so their halves are equal too.
- OA = OB = OC = OD — correct. Yes! The diagonals are equal and each is cut in half, so all four parts are equal.
- OA = OB but OC is different. OC = OA because O is the middle of AC. So OC is not different.