Rectilinear Figures [Quadrilaterals: Parallelogram, Rectangle, Rhombus, Square and Trapezium]
244. The Rectangle's Balance · Why diagonals always match in length
The diagonals AC and BD are always equal in length.
A rectangle is a parallelogram with all angles 90°. Its diagonals are equal in length (AC = BD), since each diagonal forms two congruent right triangles.
What this lesson covers
Try to break it
Drag A or C to stretch the rectangle wide or tall. The diagonals AC and BD always come out the same length and always bisect each other at O. Try to drag the corners to a setting where AC ≠ BD — impossible. Equal diagonals are the signature property of a rectangle.
How you build it
Construct a rectangle and compare its diagonals.
- Point tool: mark point A at the bottom-left.
- Point tool: mark point B to the right of A.
- Segment tool: join A to B — the base.
- Perp tool: click A (on AB), then click above — the left side, perpendicular to AB.
- Perp tool: click B (on AB), then click above — the right side, perpendicular to AB.
- Point tool: mark point D on the perpendicular at A — this sets the height of the rectangle.
- Parallel tool: click D, then click on AB — the far side, parallel to AB.
- Point tool: mark point C where the parallel line meets the perpendicular at B — rectangle ABCD is complete.
- Segment tool: join A to C — one diagonal.
- Segment tool: join B to D — the other diagonal. Measure both: AC = BD.
The proof, step by step
Prove that the diagonals of a rectangle are equal.
- AD = BC (Opposite sides of a rectangle are equal in length)
- AB = AB (Common side shared by both triangles)
- ∠DAB = ∠CBA = 90° (Each angle of a rectangle is a right angle)
- ΔDAB ≅ ΔCBA (By SAS Congruence Criterion)
- AC = BD (CPCT - Corresponding Parts of Congruent Triangles)
Worked example
In rectangle ABCD, the diagonals intersect at O. If AC = 10 cm, what is the length of BD?
By Theorem 17, the diagonals of a rectangle are equal. Since AC = 10 cm, BD must also be 10 cm.
- 5 cm
- 8 cm
- 10 cm — correct
- 12 cm