Diagonals of a rectangle

Take a rectangle PQRS and join PR and QS. These two lines, each from a corner to the opposite corner, are the diagonals of the rectangle. Predict first, then construct and measure: the two diagonals of a rectangle are always equal in length. They also cross exactly at their middles, cutting each other in half.

The right angles at P and R are called opposite angles; so are the right angles at Q and S. The diagonal PR cuts each of the opposite angles at P and R into two smaller angles. Since the whole angle is 90°, the two smaller parts always add up to 90°.

The book constructs a rectangle in which a diagonal splits the opposite angles into 60° and 30° (60 + 30 = 90). Then it asks for one with 50° and 40° (50 + 40 = 90). One way: draw the base, draw a right angle at each end (a set square is quickest), and from one end draw a line at 50° with the protractor. Where that line meets the far upright is a corner of the rectangle, and the line is its diagonal. In a long rectangle the two parts are unequal: the diagonal leans closer to the long side, so the part next to the long side is the smaller one.

When are the two parts equal? Each would be half of 90°, which is 45°. The book asks you to construct a rectangle where the diagonal splits the angles into 45° and 45°, and to look at the sides. You find that all four sides are equal: it is a square. So the diagonal splits the right angles into two equal parts only when the rectangle is a square.