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Quadrilaterals

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Quadrilaterals

A quadrilateral is a two-dimensional closed shape with four straight sides. Some of the most common types of quadrilaterals are the parallelograms, the rectangles, the squares, the trapezoids and the rhombuses. We shall look at these in detail in this chapter.


Area of a quadrilateral

Area of a quadrilateral ABCD = Area of triangle ABD + Area of triangle BCD
= (1/2)BD x d1 + (1/2)BD x d2
= (1/2)BD x (d1 + d2)

Therefore, area of a quadrilateral is equal to half the product of one of its diagonal and the sum of the perpendiculars drawn on its diagonal from the other two vertices.

Some of the common features of a quadrilateral are:
a. All quadrilateral have four sides and are coplanar.
b. Each quadrilateral must have two diagonals.
c. The sum of the four interior angles of a quadrilateral is equal to 360 degrees.

Trapezoids

A trapezoid is a quadrilateral in which only one pair of opposite sides is parallel.

Area = (1/2) x (sum of parallel sides) x (distance between them i.e. height)
If the non-parallel sides of a trapezium are equal, then it is called an isosceles trapezium. For example, in the figure below, non-parallel sides AD and BC are equal. Also, ∠A = ∠B and ∠C = ∠D.

Parallelogram

A parallelogram is a quadrilateral in which both the pairs of opposite sides are parallel.



Basic properties of a parallelogram
a. Opposite sides are parallel and equal.
b. Opposite angles are congruent.
c. Adjacent angles are supplementary
d. Diagonals bisect each other
e. Each diagonal divides the parallelogram into two congruent triangles
f. Diagonals divide the parallelogram into four triangles of equal area.

Area of a parallelogram
Area = base x height (or)
Area = Product of two sides x Sine of included angle

Rectangles
A rectangle is a quadrilateral whose opposite sides are parallel and whose interior angles are all right angles. A rectangle is essentially a parallelogram whose angles are all right angles.


Basic properties of a rectangle
a. The diagonals are equal.
b. Diagonals bisect each other.
c. Each angle is a right angle.
d. Each diagonal divides the rectangle into two congruent triangles.
e. Diagonals divide the rectangle into four triangles of equal area.

Area of a rectangle
Area = length x breadth

Rhombus
A rhombus is a quadrilateral whose opposite sides are parallel and whose sides are of equal length.



Area of a rhombus
Area = (1/2) x product of the diagonals

Square
A square is a quadrilateral in which all the sides are equal and all the angles are right angles.


Area of a square
Area = (side)2



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