What are the properties of the diagonals of a rectangle?

Answer 1

A Rectangle is a quadrilateral (four sided polygon) with two pairs of equal and parallel sides (opposite sides are parallel and equal, one pair is usually a different length from the other pair but if they are equal it is called a square), and all angles are right angles (90°). It has two diagonals which have the properties:

1. The diagonals are always congruent (of equal length);
2. The diagonals bisect each other (cut each other into two equal parts);
3. The diagonals do not bisect the angles (unless the rectangle is a square when they do);
4. The diagonals are not perpendicular (unless the rectangle is a square when they do).

PROOF of the diagonals congruent:

Take a rectangle ABCD with diagonals AC and BD.

Using Pythagoras on the triangles ACD and BCD:

AC² = AD² + CD²

BD² = BC² + CD²

But as ABCD is a rectangle AD = BC since they are opposite and parallel; thus:

AC² = AD² + CD² = BC² + CD² = BD²

Thus, as AC and BD are the diagonals, they are equal.

Therefore the diagonals of a rectangle are congruent.

Answer 2

True. This can be deduced from the hypotenuse of similar triangles.

Answer 3

True.