These notes for CBSE Class 9 Mathematics cover Chapter 9: Areas of Parallelograms and Triangles. The chapter begins by defining a planar region and its area, and introduces the concept of congruent figures, emphasizing that while congruent figures have equal areas, figures with equal areas are not necessarily congruent. It then explains figures on the same base and between the same parallels, defining them as figures sharing a common base with opposite vertices lying on a line parallel to the base. Key theorems are presented: parallelograms on the same base and between the same parallels have equal areas, and a triangle on the same base and between the same parallels as a parallelogram has half its area. The notes also state that two triangles on the same base and between the same parallels are equal in area. Formulas for the area of a parallelogram (base x altitude) and a triangle (1/2 x base x altitude) are provided. These notes are ideal for quick revision before exams.
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You may recall that the part of the plane enclosed by a simple figure is called a planar region
corresponding to that figure. The magnitude or measure of this lanar region is called its area.
We are also familiar with the concept of congruent figures.
Two figures are called congruent, if they have the same shape and the same size.
So if two figures A and B are congruent they must have equal areas. However, the converse of
this statement is not true. In other words, two figures having equal areas need not be congruent.
For example, in figure, rectangles ABCD and EFGH have equal areas (9 x 4 cm2 and 6 x 6
cm2) but clearly they are not congruent.
FIGURES ON THE SAME BASE AND BETWEEN THE SAME PARALLELS
Two figures are said to be on the same ‘base add between the same parallels, if they have a
common base (side) and the vertices (or the vertex) opposite to the common base of each figure
lie on a line parallel to the base.
If a parallelogram PQRS, a rectangle EFGH and a triangle LMN are so drawn that their bases
lie on a straight line (say, CD) and their other vertices lie on a line AB parallel to it (CD), then
the parallelogram, the rectangle and the triangle are said to be between the same parallels.
Note : Out of the two parallels, one must be the line containing the common base. ?ABC and
?DBE of figure are not on the common base.
Statement : Parallelograms on the same base and between the same parallels are equal in area.
Remark: Area of a parallelogram is the product of its any side and the corresponding altitude.
Statement : if a triangle an a a parallelog Ain Are mit* same base and between the same
parallels, then the area of the triangle is equal to half the area of the parallelogram.
Two triangles on the same base (or equal bases) and between the same parallels are equal in
area.
Note : Area of ?ADC = 1/2 x base DC x corresponding altitude AN
In other words, area of a triangle is half the product of its base (or any side) and the
corresponding altitude (or height).
Remarks : Two triangles with same base (or equal bases) and equal areas will have equal
corresponding altitudes.
Theorem : Two triangles having the same base (or equal bases) and equal areas lie between the
same parallels.
The area of a figure is the measure of the planar region enclosed by it. Two figures are on the same base and between the same parallels if they share a common base and their vertices opposite to the common base lie on a line parallel to the base. Yes, parallelograms on the same base and between the same parallels are equal in area. If a triangle and a parallelogram are on the same base and between the same parallels, the area of the triangle is half the area of the parallelogram. Yes, two triangles on the same base (or equal bases) and between the same parallels are equal in area. The area of a parallelogram is the product of its base and the corresponding altitude. The area of a triangle is half the product of its base and the corresponding altitude.
Frequently asked questions
What is the definition of area in mathematics?
What does it mean for two figures to be on the same base and between the same parallels?
Are parallelograms on the same base and between the same parallels equal in area?
What is the relationship between the area of a triangle and a parallelogram on the same base and between the same parallels?
Are two triangles on the same base and between the same parallels equal in area?
What is the formula for the area of a parallelogram?
What is the formula for the area of a triangle?
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