CBSE Class 9 Mathematics Areas of Parallelograms and Triangles Notes Set 19

Area and Congruent Figures

The area of a plane region is the measure of the part of the plane it encloses, in square units.
Area axioms: (1) every region has an area; (2) congruent figures have equal areas; (3) if a region is made of two non-overlapping regions, its area is the sum of their areas.
The converse of (2) is false: a 9 cm by 4 cm rectangle and a 6 cm by 6 cm square both have area 36 cm², but they are not congruent.

Parallelograms on the Same Base and Between the Same Parallels

A diagonal divides a parallelogram into two triangles of equal area.
Parallelograms (and a parallelogram and a rectangle) on the same base and between the same parallels have equal areas.
Area of a parallelogram = base × corresponding altitude. Parallelograms on equal bases and between the same parallels also have equal areas.
Of all segments from a point to a line, the perpendicular is the shortest, so a parallelogram has a greater perimeter than a rectangle of the same base and area.

Triangles on the Same Base and Between the Same Parallels

Area of a triangle = ½ × base × corresponding altitude.
Triangles on the same base (or equal bases) and between the same parallels have equal areas.
Converse: triangles with the same base and equal areas 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: the triangle has half the area.
A median divides a triangle into two triangles of equal area; the three medians divide it into six triangles of equal area.

Area of Similar Triangles

In similar triangles the ratio of corresponding sides is the same.
The ratio of their areas is the square of the ratio of corresponding sides: ar(ABC) ÷ ar(PQR) = (AB/PQ)² = (BC/QR)² = (AC/PR)².
Examples: the mid-point triangle has 1/4 of the area; a line parallel to a side of a triangle cuts off a similar triangle whose area ratio is the square of the side ratio.
The area of an equilateral triangle of side a is (√3/4)a².

Worked Examples

Example 1: If each diagonal of a quadrilateral divides it into two triangles of equal area, the quadrilateral is a parallelogram. Each of ar(ABC) and ar(ABD) is half the area of ABCD, so they are equal; being on the same base AB they have equal altitudes and DC ∥ AB. In the same way AD ∥ BC.
Example 2: Parallelograms PQRS and PABC have equal areas (P, Q, R, S and A, B, C as in the usual figure). Joining BQ, QC, CR, RB and subtracting the common part shows ar(△BRC) = ar(△BRQ). They are on the same base BR, so they lie between the same parallels and QC ∥ BR.
Example 3: A parallelogram ABCD and a rectangle ABEF on the same base AB have equal areas. Then AB + CD = AB + EF. Because BE < BC and AF < AD (perpendicular is the shortest), the perimeter of the parallelogram is greater than that of the rectangle.
Example 4: In a parallelogram, one altitude is 7 cm for the 10 cm side and the other altitude is 8 cm. Area = 10 × 7 = 70 cm², so the other side = 70 ÷ 8 = 8.75 cm.