Quadrilateral

1. Introduction To Quadrilateral

A quadrilateral is considered to be referred to as the four-sided polygon which is supposed to have four angles as these serve as the types of quadrilaterals which are considered to be parallelogram that acts as rhombus or kite along with being rectangle as well as a trapezoid, square and along with such isosceles trapezoid. It is used in geometry and it is defined by a two-dimensional shape that has four straight sides.

2. Various Types of Quadrilaterals

A. The Rectangle

The quadrilateral of a rectangle is supposed to be through the four right angles where the angles are supposed to be equal to (360 degrees/4as such equals 90 degrees). However, the opposite sides of a rectangle are supposed to be parallel and along with such equal as the diagonals bisect each other.

B. The Square

A square is supposed to be a regular quadrilateral which implies that it has four equal sides whereby there are four angles. Therefore, these are supposed to be adjacent to having an equal length. Therefore, a square with vertices ABCD would be denoted and demonstrated as ABCD. The only quadrilateral of a square is supposed to be through the regular where all the sides are considered to be equal along with the angles as such is the quadrilateral of a square which makes all the other irregular.

C. The Rhombus

A rhombus is supposed to be a quadrilateral whose four sides are supposed to be having the same length and this has another name which relates to equilateral quadrilateral and such is regular. There are three approaches where the first approach would be showing or depicting the shape and such would have to be a parallelogram which would be having equal length sides. In addition to this, the shape’s diagonals need to be paired with each other’s’ perpendicular bisectors and such needs to be shown in shape through the diagonals where the bisection of both the pairs would be through the opposite angles.

D. The Parallelogram

It is supposed to be a simple non-self-intersecting quadrilateral which is supposed to have two pairs of parallel sides and the quadrilaterals are supposed to be with the two sets of parallel sides and due to such the squares need to be met with as the opposite sides need to be congruent along with such being in consecutive angles which are supposed to be supplementary as one of the angles are supposed to be right as these are diagonals through the parallelogram as such is supposed to bisect each other.

E. The Trapezoid

The quadrilateral of a trapezoid is supposed to be considered to be having only one pair of the parallel sides as these exclude parallelogram. However, it is also defined as having the least sides which makes the parallelogram a special kind of trapezoid.

F. The Kite

It is a quadrilateral whose four sides are supposed to be grouped through the two pairs which consist of equal length as these are supposed to be adjacent to each other. However, in contrast these also have two pairs for the equal-length sides which are opposite to each other as such are not adjacent.

3. Convex, Concave, and Intersecting Quadrilaterals

A convex is supposed to be a quadrilateral that has both the diagonals completely contained within a figure and on the other hand, concave is supposed to have at least one diagonal that would be lying partly or entirely outside of the figure. It can be understood through the formula “(n − 2) × 180°” as these are supposed to be through an interior angle sum formula. All the other non-self-crossing quadrilaterals are supposed to help in tiling the planes which would be repeated through rotation around the midpoints as these would be through those ages. The intersecting quadrilaterals are supposed to not be simple as these would be made from a pair of non-adjacent sides as such are supposed to intersect. Therefore, these kinds of quadrilaterals are supposed to be known as self-intersecting or crossed quadrilaterals.

4. Venn Diagram of Quadrilateral Classification

It can be stated that the quadrilateral is supposed to be a closed figure. Thus, it is supposed to be a flat geometric figure which has four sides as well as four angles.

5. Properties of Quadrilateral

The quadrilateral is supposed to be closed shapes which would be having all the internal angles sum up to three-hundred-and-sixty degrees. The opposite angles need to be equal and along with such all the sides would have to be equal where the opposite sides would be parallel to each other. The diagonals are also supposed to bisect each other in a perpendicular manner and the sum of the two angles would have to be 180 degrees.

6. Area of Quadrilaterals

“(½) × diagonal length × sum of the length of the perpendiculars drawn from the remaining two vertices.” This implies that the area of the quadrilateral would have to be found by dividing into two triangles as such would be using a diagonal. These would also be dependent upon the length as well as the height of the diagonal as there are two triangles that would be known which would be based on the area of the quadrilateral.

7. Top Quadrilateral Examples

Example 1:

“A rhombus has diagonals of 12 and 6 units. Find its area.

Solution:

Given, d1 = 12, and d2 = 6

To find: area

Formula: 1/2 (d1 × d2)

Area of = 1/2 (d1 × d2)

Area =1/2 (12 x 6)

Area = 36

Area of the rhombus is 36 square units”.

Example 2:

“Calculate the area of the quadrilateral formed with the vertices (−3, 2), (5, 4), (7, −6) and (−5, −4).

Solution:

Let A(-3, 2), B(5, 4), C(7, -6) and D(-5, -4) be the vertices of a quadrilateral ABCD.

Thus,

A(-3, 2) = (x1, y1)

B(5, 4) = (x2, y2)

C(7, -6) = (x3, y3)

D(-5, -4) = (x4, y4)

We know that,

Area of quadrilateral ABCD = (1/2) ⋅ [(x1y2 + x2y3 + x3y4 + x4y1) – (x2y1 + x3y2 + x4y3 + x1y4)]

Substituting the values,

= (½). {[-3(4) + 5(-6) + 7(-4) + (-5)2] – {[5(2) + 7(4) + (-5)(-6) + (-3)(-4)]}

= (½).[(-12 – 30 – 28 – 10) – (10 + 28 + 30 + 12)]

= (½) [-80 – 80]

= 160/2 {since area cannot be negative}

= 80

Therefore, the area of the quadrilateral formed with the given vertices is 80 sq. units.”

Example 3:

“Find the area of a rhombus whose diagonals are 7 cm and 6 cm respectively.

Solution:

Given: Diagonal 1, p = 7 cm

Diagonal 2, q = 6 cm.

The area of rhombus = (½)×d1×d2 square units.

A = (½)(7)(6) cm2

A = 7(3) cm2

A = 21 cm2.

Therefore, the area of a rhombus is 21 cm2.”