Area of Parallelogram and Triangle Class 9 Notes
Introduction
Polygonal region
The part of the plane enclosed by a polygon is called the interior of a polygon and the union of a polygon and its interior is called a polygonal region.
If the polygon is triangle then it is called triangular region.
Figure
Important Axioms
(1) Area axiom : Every polygonal region has an area, measured in square units. The area of a polygonal region R is denoted by “ar (R)”.
(2) Congruent area axiom : If ΔABC and ΔDEF are two congruent triangles, then ar(ΔABC) = ar(ΔDEF) i.e. two congruent regions have equal area.
(3) Area monotone axiom : If R1 and R2 are two polygonal regions such that R1 is a part of R2, then ar (R1) \le ar (R2).
(4) Area addition axiom : If a plane region formed by a figure R is made up of two non overlaping plane regions formed by R1 and R2 then ar(R) = ar(R1) + ar(R2).
Figure
(5). Rectangle area axiom : If ABCD is a rectangular region such that AB = a units and BC = b units, then ar (ABCD) = ab sq. units.
(6) Figures on the same base and between the same parallel lines : Two geometric figures are said to be on the same base and between the same parallel lines, if they have a common side (base) and the vertices opposite to the common base of each figure lie on a line parallel to base.
Figure
In figure (I) quadrilateral ABCD and ΔABE are on same base AB and lie between same parallel lines AB and DC.
In figure (II) quadrilateral PQRS and quadrilateral PQNM are on same base PQ but does not lie between same parallels because vertices M andN of quadrilateral PQNM and R & S of quadrilateral PQRS do not lie on the same line.
In figure (III) quadrilateral ABCD and ΔAEF lie between same parallels AB and DC but they do not have a common base.
Important Theorems
Theorem 1 A diagonal of a parallelogram divides it into two triangles of equal area.
In a parallelogram ABCD in which BD is one of the diagonals.
Then, ar (ΔABD) = ar (ΔCDB)
Figure
Theorem 2 Parallelograms on the same base and between the same parallels are equal in area.
In two parallelograms ABCD and ABEF, which have the same base AB and are between the same parallel lines AB and FC.
Then, ar (||gm ABCD) = ar (||gm ABEF)
Figure
Theorem 3 Parallelograms on equal bases and between the same parallels are equal in area.
Two parallelograms ABCD and PQRS with equal bases AB and PQ and between the same parallels AQ and DR.
Figure
Then, ar (||gm ABCD) = ar (||gm PQRS)
Theorem 4 If a triangle and a parallelogram are on the same base and between the same parallels, the area of the triangle is equal to half of the parallelogram.
If a ΔABC and a ||gm BCDE on the same base BC
and between the same parallels BC and AD.
Then, ar (ΔABC) = \frac{1}{2} ar (||gm BCDE)
Triangles on same base and between same parallels
Two triangles are said to be on the same base and between same parallel lines, if they have a common base and vertices opposite to the common base of each triangle lie on a line parallel to the common base.
For example : In given figure Δ ABC and Δ ABD are on the same base AB and between same parallels AB and CD
Figure
Important Theorem
Theorem 1 : Triangles on the same base and between the same parallels are equal in area.
Two ΔsABC and DBC on the same base BC and between the same parallel lines BC and AD.
Then, ar (ΔABC) = ar. (ΔDBC)
Figure
Theorem 2 : If two triangles have equal areas and one side of the one triangle is equal to one side of the other then their corresponding altitudes are equal.
Two Δs ABC and DEF such that ar. (ΔABC) = ar. (ΔDEF) and BC = EF.
Figure (i) Figure (ii)
Then altitude AL = altitude DM
Theorem 3
A median of a triangle divides it into two triangles of equal area.
A ΔABC in which AD is the median.
Then, ar. (ΔABD) = ar. (ΔACD)
Figure