The Diagonals of a Rectangle

Imagine making a rectangular frame without starting from its four edges. You have one wooden strip 8 cm long, another strip you can cut to size, and a thread to stretch around their four endpoints. The strips will run across the frame as diagonals. Where should you join them? Must they form a right angle? Thinking about this practical problem will help you discover a second way to recognise and construct a rectangle.

We will use congruent triangles: triangles that match exactly in size and shape. SAS means two sides and their included angle match; ASA or AAS uses two matching angles and a corresponding side. Always write triangle names so matching vertices occupy matching positions.

A Carpenter’s Problem

Let ABCD be a rectangle, named around its boundary, and let diagonals AC and BD meet at O. We will first start with a rectangle and find what its diagonals must do. Then we will reverse the reasoning and construct a rectangle from those conditions.

The three questions are the length of the second strip, where the strips meet and the angle at the joint. Do not assume the joint is a right angle merely because the frame has right-angled corners.

Deduction 1 — What is the length of the other diagonal?

Compare triangles BAD and CDA. BA = CD because they are opposite rectangle sides. AD is common. Their included angles ∠BAD and ∠CDA are both 90°.

The triangles are congruent by SAS, with B ↔ C, A ↔ D and D ↔ A. Therefore their corresponding third sides BD and CA are equal. An 8 cm first diagonal requires an 8 cm second diagonal.

Deduction 2 — What is the point of intersection of the two diagonals?

Compare triangles AOB and COD. Their angles at O are vertically opposite, so they are equal. To compare another pair, notice that ∠ABO + ∠OBC = 90°. In triangle BCD, ∠OBC + ∠CDO + 90° = 180°. Hence ∠ABO and ∠CDO are both 90° − ∠OBC.

AB = CD. Together with the two matching angles, this gives triangle AOB congruent to triangle COD by AAS. Thus AO = CO and BO = DO. The point O is the midpoint of both diagonals.

Deduction 3 — What are the angles between the diagonals?

Now take two equal strips and join them at their midpoints. All four half-strips have the same length. Suppose ∠AOB = 60°. Its vertically opposite angle is 60°, and the two neighbouring angles at O are 120°.

Triangle AOB is isosceles. Its base angles are (180° − 60°) ÷ 2 = 60°. Triangle AOD is also isosceles, with base angles (180° − 120°) ÷ 2 = 30°. At A, the two parts add to 60° + 30° = 90°. The same happens at every corner.

Replace 60° with any angle x between 0° and 180°. The neighbouring central angle is 180° − x. The first base angle is 90° − x/2, and the second is x/2. Adding them gives 90°.

Opposite outer triangles are congruent by SAS: two equal half-strips and equal vertically opposite angles. Consequently, opposite boundary sides are equal. The quadrilateral is therefore a rectangle. Avoid 0° and 180°, when the endpoints collapse onto one line.

The Process of Finding Properties

Measuring several rectangles may suggest that their diagonals always bisect each other. This is useful evidence for a conjecture: a statement you think is true. It does not check every possible rectangle, and every measurement has some uncertainty.

A proof starts from defining conditions and already established facts. The congruence arguments above do not depend on a particular width or height. They explain why the property holds for every rectangle.

Deduction 4 — What is the shape of a quadrilateral with all the angles equal to 90°?

Suppose only the four right angles are given. Draw diagonal BD. Using the same complementary-angle reasoning as above, ∠ABD = ∠CDB. Also ∠BAD = ∠DCB = 90°, and BD is shared.

Triangles BAD and DCB are congruent by AAS. Hence BA = DC and AD = CB. The side equality follows from the angle condition; it does not need to be a separate requirement.

The opposite sides are parallel too. For example, the same-side interior angles at A and B sum to 180°, so AD ∥ BC. Applying the same converse parallel-line test to another pair gives AB ∥ CD.