Diagonals of Rectangles and Squares

What is a Diagonal?

A diagonal is a line segment connecting two opposite corners of a polygon. In a rectangle $PQRS$, the lines $PR$ and $QS$ are diagonals.

Properties

1. Equal Lengths: In both squares and rectangles, the two diagonals are equal in length ($PR = QS$).
2. Bisecting: They cut each other exactly in half.
3. Square Speciality: In a square, the diagonals intersect at right angles ($90^\circ$).

Constructing with Diagonals

Problem: Construct a rectangle where side $CD = 5$ cm and diagonal $BD = 7$ cm.

Logic:

1. Draw the base $CD = 5$ cm.
2. We know the angle at $C$ is $90^\circ$. Draw a perpendicular line at $C$. Let’s call this line $l$.
3. Point $B$ lies on line $l$.
4. Point $B$ is also 7 cm away from $D$.
5. Using a compass, set radius to 7 cm, place needle at $D$, and cut an arc on line $l$.
6. The intersection is point $B$.
7. Complete the rectangle.