QUADRILATERALS


A closed figure (plane) bounded by four sides is called a quadrilateral. In any quadrilateral :

\(A = \frac{1}{2} \times\) one diagonal \(\times\) (sum of perpendiculars to it from opposite vertices).

(a) Sum of the four interior angles = 360°
(b) If a quadrilateral can be inscribed in a circle, it is called a cyclic quadrilateral. Here opposite angles are supplementary. If one side is produced, then the exterior angle = Remote interior angle.


Rectangle



A quadrilateral whose opposite sides are equal and each internal angle equal to 90°, is called a rectangle.

\(l = length\)
\(b = breadth\)
\(A = l \times b\)
\(P = 2 (l + b)\)

Diagonal \(= \sqrt{(l^2 + b^2)}\), diagonals are equal and bisect each other.

(a) Opposite sides equal, each angle = 90°
(b) Diagonals bisect each other (not at 90°).
(c) Of all rectangles of given perimeter, a square has max. area
(d) When inscribed in a circle, it will have maximum area when it’s a square.
(e) Figure formed by joining the midpoints of a rectangle is a rhombus.
(f) The biggest circle that can be inscribed in a rectangle will have the diameter equal to the breadth of the rectangle.



(g) When a rectangle is inscribed in a circle, the diameter of the circle is equal to the diagonal of the rectangle.



Parallelogram




A quadrilateral in which opposite sides are equal and parallel is called a parallelogram.

\(A = bh\)
\(P = 2 (a + b)\), Diagonals bisect each other.

(a) Opposite angles are equal
(b) Diagonals bisect each other (not at 90°)
(c) Sum of any two adjacent angles = 180°
(d) Diagonals are unequal in lengths and do not bisect angles at vertices.



Square



A quadrilateral whose all sides are equal and all angles 90° is called a square

\(Area=a^2 \)

\(Diagonal=a\sqrt{ 2} \)

\(Area=\frac{Diagonal^2}{2}\)

\(P=4 \times side\)


(a) Diagonals bisect each other at 90° and are equal
(b) When inscribed in a circle, diagonal = diameter of circle
(c) When circumscribed about a circle, Side of square = Diameter of circle.
(d) The figure formed by joining the mid-points of the sides of a square is also a square. In this case the side will become \(\frac{1}{\sqrt{2}}\) times the side of the original square, perimeter will become \(\frac{1}{\sqrt{2}}\) times the perimeter of the original square and area will become \(\frac{1}{2}\) times the area of the original square.
(e) If a square (biggest possible) is inscribed in a circle of radius r, then Diameter of circle = Diagonal of Square. If the area of the circle is A then \(A = \pi r^2=\frac{\pi D^2}{4} or \frac{D^2}{2} =\frac{2A}{\pi}\) or Area of Square \(= \frac{2A}{\pi} or 2r^2\)
(f) If a circle is inscribed in a square of side a, then Side of square = Diameter of circle. If the area of the square is S, then the area of circle \(= \frac{\pi S}{4} =\frac{ \pi a^2}{4}\)



Rhombus

A parallelogram whose all sides are equal is called a rhombus.




\(A=\frac{1}{2} \times D_1 \times D_2 ( D_1, D_2 are diagonals)\)

\(P=4a\)

Diagonals are at right angles

\(D_1^2+D_2^2=4a^2\)

(a) Opposite angles are equal
(b) Diagonals bisect each other at 90°.
(c) Sum of any two adjacent angles = 180°.
(d) Diagonals are unequal.



Trapezium



A quadrilateral in which one pair of opposite sides is parallel is a trapezium. \(A =\frac{1}{2} \times (a + b) \times h\)



CYCLIC QUADRILATERAL




1. The four vertices lie on a circle.
2. Opposite angles are supplementary.
3. If any one side is produced, Exterior angle = Remote interior angle