9-Areas of Parallelograms and Triangles

Areas of Parallelograms

We know that Area of a parallelogram is equal to the product of its base and height.
The diagonals of a parallelogram bisect each other.
A diagonal of the parallelogram divides it into two congruent triangles.
Opposite sides and angles of a parallelogram are equal.
Parallelograms on the same base and between the same parallels are equal in area.
Parallelograms on the same base that have equal areas are between the same parallels.
All figures that are on the same base and between same parallels need not have equal areas.
Two congruent figures have equal areas but the converse need not be true.
Parallelograms on the same base or equal bases and lie between the same parallels are equal in area.
Parallelograms on the same base or equal bases that have equal areas lie between the same parallels.
If a parallelogram and a triangle are on the same base and they are between the same parallels, then area of the triangle is half the area of the parallelogram

 

 

 

 

 

 

Corollary: Parallelograms on the same base that have equal areas are between the same parallels.

Proof: ar (ABCD) = ar (ABEF)…eq (1)

Area of parallelogram = b x h

b x h₁=b x h₂

h₁=h₂

Thus, the corollary is proved.

 

 

 

 

 

 

Areas of Triangles

Two figures are said to be on the same base and between the same parallel lines if they have a common base and the vertices opposite to this common base of each figure lie on a line parallel to the base. Out of these two parallel lines, one should be the line containing the common base and the other is the line passing through the vertices of both the figures opposite to the base. We know that any two congruent figures have same area but the converse is not true. Median of a triangle divides it into two triangles of equal area. Two triangles with same base (or equal bases) and equal areas will have equal corresponding altitudes.

Area of triangle is half the product of the base and the corresponding altitude. Also, Triangles on the same base and between the same parallels are equal in area.

Triangles on the same base and having equal areas lie between the same parallels.

The Area of a triangle is half the product of the base and the corresponding altitude.

Theorem: Triangles on the same base and between the same parallels are equal in area.