Theorem

The diagonals of a rectangle are equal and bisect each other

Geometry • Quadrilateral • Parallelogram

Statement

The diagonals of any parallelogram bisect each other; in a rectangle they are in addition equal, so their common midpoint is equidistant from all four vertices, while in a rhombus they are instead perpendicular, and in a square both extra properties hold at once.

How to use it

1Half a diagonal of a rectangle is the distance from the centre to every vertex, so get the diagonal by Pythagoras and halve it.
2Use the equality of the diagonals whenever a construction turns half-diagonals into lengths, as when joining the midpoints of the sides.
3Read the converse as a test: equal diagonals single out the rectangle among parallelograms, perpendicular diagonals the rhombus.

Worked example

A rectangle has sides

6

cm and

8

cm, and its diagonals meet at

O

. How far is

O

from each vertex?

Step-by-step

1
diagonal $= 6^{2} + 8^{2} = 100 = 10$ cm
Why: Two sides and a diagonal form a right triangle.
2
$O$ bisects each diagonal, and the diagonals are equal, so every vertex is $\frac{10}{2} = 5$ cm from $O$

Answer

5

Watch out

Bisection alone does not give equality: a non-rectangular parallelogram has diagonals that bisect each other but differ in length. Never read equality or perpendicularity of diagonals off a drawing that does not mark it.

Builds on

Pythagoras' theorem

Also known as

diagonals of a rectangle are equal, diagonals bisect each other, properties of a rectangle, diagonals of a parallelogram, diagonals of a rhombus are perpendicular, half the diagonal

Appears in past papers

Statement

Worked example

Appears in past papers

2020 (I)

2019 (II)

2018 (I)

2017 (II)

2017 (I)