Introduction of Areas of Parallelograms and Triangles Notes

Introduction

The area represents the amount of planar surface being covered by a closed geometric figure.

Areas of closed figures

Figures on the Common Base and Between the Same Parallels

Two shapes are stated to be on the common base and between the same parallels if:
a) They have a common side.
b) The sides parallel to the common base and vertices opposite the common side lie on the same straight line parallel to the base.

For example Parallelogram  ABCD, Rectangle ABEF and Triangles ABP and ABQ

Area of a parallelogram

Parallelogram

Area of a parallelogram = b×h

Where ‘b′ is the base and ‘h′ is the corresponding altitude(Height).

Area of a triangle

Area of triangle

Area of a triangle = (1/2)×b×h

Where “b” is the base and “h” is the corresponding altitude.

Theorems

Parallelograms on the Common Base and Between the Same Parallels

Two parallelograms are said to be on the common/same base and between the same parallels if
a) They have a common side.
b) The sides parallel to the common side lie on the same straight line.

Parallelogram ABCD and ABEF

Theorem: Parallelograms that lie on the common base and between the same parallels are said to have equal in area.
Here, ar(parallelogram ABCD)=ar(parallelogram ABEF)

Triangles on the Common Base and Between the Same Parallels

Two triangles are said to be on the common base and between the same parallels if
a) They have a common side.
b) The vertices opposite the common side lie on a straight line parallel to the common side.

Triangles ABC and ABD

Theorem: Triangles that lie on the same or the common base and also between the same parallels are said to have an equal area.
Here, ar(ΔABC)=ar(ΔABD)

Two Triangles Having the Common Base & Equal Areas

If two triangles have equal bases and are equal in area, then their corresponding altitudes are equal.

A Parallelogram and a Triangle Between the Same parallels

A triangle and a parallelogram are said to be on the same base and between the same parallels if
a) They have a common side.
b) The vertices opposite the common side lie on a straight line parallel to the common side.

A triangle ABC and a parallelogram ABDE

Theorem: If a triangle and a parallelogram are on the common base and between the same parallels, then the area of the triangle is equal to half the area of the parallelogram.
Here,  ar(ΔABC)=(1/2) ar(parallelogarm ABDE)