Understanding Quadrilaterals

Quadrilaterals

A quadrilateral is any closed plane figure with four sides and four vertices. There are several different shapes that can fit this description, each with its own distinct characteristics and applications.

Types of Quadrilaterals

The most common types of quadrilaterals are squares, rectangles, rhombuses, and parallelograms:

Squares

Squares are special quadrilaterals in which all sides are equal. They have exactly two pairs of opposite sides congruent to one another. A square has four right angles, and its corners are all 90 degrees apart. All squares are also rectangle and they are the only quadrilateral whose interior angles sum up to 540°. Regular squares can be constructed by using the regular polygon construction theorem. For example, if the length of each side of a square is x units, then the square's perimeter is x + x + x + x = 4x units. The area of a square is x * x, since every unit area is covered twice over. This implies that a square is a rectangle with both lengths equal. This leads us to the definition of a square as a rectangle with equal sides. Thus, a square is a rectangle composed of four equilateral triangles.

Rectangles

Rectangles are quadrilaterals where adjacent sides are pairwise congruent, meaning opposite sides are parallel. Unlike squares, their sides are not necessarily equal, so there is more than one way to partition a rectangle into congruent squares. One way to construct a rectangle with perpendicular sides is to start with two lines that are parallel to the same line, then draw a perpendicular line through one endpoint of those lines. Then draw another line from the other endpoint of the original lines to meet the first perpendicular line.

A rectangle has four right angles, and its corners are all 90 degrees apart. Opposite angles of a rectangle are congruent. Given a rectangle with base b and height h, the area of the rectangle is calculated as the product of its base and height (b * h), while its perimeter is given by the formula P = 2l + 2w, where l represents the length and w represents the width of the rectangle. Additionally, a rectangle is also defined as a square truncated by four lines parallel to the edges of the square. In terms of trigonometry, a rectangle is a quadrilateral in which all internal angles are either 90° or less, i.e., it contains no obtuse angles. It follows that the sum of the internal angles of a rectangle is always 360°.

Rhombuses

Rhombuses are quadrilaterals with equal opposite sides, which means that opposite sides of a rhombus are parallel to each other. If a rhombus has side of length s, then the distance between non-consecutive vertices is s√2. The area of a rhombus can be found by multiplying half the length of a diagonal by the altitude drawn from the vertex to the opposite side, which intersects the angle bisector. However, unlike squares and rectangles, there is no simple formula to find the area of a generalized rhombus when we know the measure of only three of its sides.

Parallelograms

Parallelograms are quadrilaterals that have opposite sides parallel and congruent. While not all parallelograms have four right angles, the diagonals of a parallelogram do bisect each other, meaning they divide the parallelogram into two congruent smaller parallelograms. These two pairs of congruent sides form bases of two parallelograms. Each of these parallelograms contributes half of the total area of the larger parallelogram. Therefore, the area of a parallelogram is the sum of the areas of its two congruent bases. Alternatively, if you prefer to work in terms of heights, you can think of the area of the parallelogram as the product of the height and the distance between the bases.

Properties of Quadrilaterals

Quadrilaterals share some fundamental properties due to their structure:

1. The sum of the measures of all angles in a quadrilateral is 360°.
2. Quadrilaterals have four sides and four angles.
3. Only squares and rectangles have all sides equal.
4. A parallelogram is a quadrilateral with opposite sides parallel.
5. A kite is a quadrilateral with a single axis of symmetry.
6. Two concurrent diagonals (lines that cross) divide a quadrilateral into four triangles.

Calculating the Area of a Quadrilateral

Each type of quadrilateral has a specific method for calculating its area:

• Rectangle: Multiply the length (base) by the width (height).
• Square: Square the side length.
• Parallelogram: Half the product of the base and height, or the product of the height and the distance between the bases.
• Rhombus: One-half the product of the length of a diagonal by the height.

Diagonals in Quadrilaterals

Diagonals serve various purposes within the context of a quadrilateral:

1. They often reveal whether a quadrilateral is convex or concave. Since two sets of parallel sides must lie completely to the left or right of a common perpendicular line, two diagonals cannot both be completely outside a common set of parallel sides. Hence, one of the diagonals must go inside the quadrilateral, making it a concave quadrilateral.
2. They can help determine if a quadrilateral is convex or not. If a quadrilateral is convex, the endpoints of two diagonals will alternate along the boundary. Convexity refers to the property of a shape where none of the internal points are located behind the others. More formally, if the two endpoints are connected by a straight line, then the shape is not convex.
3. In a quadrilateral with equal lateral sides, the intersection of the two diagonals lies on the midpoint of each side, dividing the quadrilateral into four equivalent regions.