CBSE Class 9 Mathematics Areas of Parallelograms and Triangles Notes Set 22

Area Axioms

Congruent figures have equal areas. If one region lies inside another, its area is not more than the other. If a region is split into parts that overlap only along segments or points, its area is the sum of the parts. A rectangle of sides a and b has area ab. Area is measured in square units.

Figures on the Same Base and Between the Same Parallels

Two figures are on the same base and between the same parallels when they share a base and their opposite vertices lie on a line parallel to the base. Such figures have the same height.

Parallelograms

A diagonal divides a parallelogram into two triangles of equal area. Parallelograms on the same base (or equal bases) and between the same parallels have equal areas. The area of a parallelogram is base × corresponding height. A rectangle and a parallelogram on the same base and between the same parallels have equal areas, so the perimeter of the parallelogram is greater than that of the rectangle.

Triangles

Triangles on the same base (or equal bases) and between the same parallels have equal areas. Conversely, equal triangles on the same base and on the same side of it lie between the same parallels. A triangle and a parallelogram on the same base and between the same parallels satisfy ar(triangle) = ½ ar(parallelogram). Area of a triangle = ½ × base × height.

Useful Results

A median divides a triangle into two triangles of equal area. If D is any point on BC, ar(△ABD) : ar(△ACD) = BD : DC. The three medians divide a triangle into six equal-area triangles. The midpoints of the sides divide a triangle into four equal triangles. Area of a trapezium = ½ × (sum of parallel sides) × height. Area of a rhombus = ½ × product of the diagonals. Area of an equilateral triangle of side a = (√3/4)a².

Solved Example 1

In a parallelogram ABCD, AB = 16 cm, the altitude on DC is AE = 8 cm and the altitude on AD is CF = 10 cm. Find AD.
Area = DC × AE = 16 × 8 = 128 cm². Also area = AD × CF = AD × 10, so AD = 12.8 cm.

Solved Example 2

In △ABC, D is the midpoint of BC and E is the midpoint of AD. Show that ar(△BED) = ¼ ar(△ABC).
BE is a median of △ABD, so ar(△BED) = ½ ar(△ABD). AD is a median of △ABC, so ar(△ABD) = ½ ar(△ABC). Hence ar(△BED) = ¼ ar(△ABC).