In the previous topics, we have learnt how to find the area of planar regions. And, also familiar with the topic "Congruence of triangles". Let us recall them.
Consider \(2\) figures \(PQRS\) and \(ABCD\), whose areas are \((4 \times 4) \ cm^2\) and \((8 \times 2) \ cm^2\). We know the statement that "Two figures are said to be congruent if they have the same shape and the same size". That is, we can also say that "Two figures are congruent if their areas are equal". But, the converse of this statement is not true. That is, "Two figures having equal areas need not be congruent".
We can see that the areas of the figures \(PQRS\) and \(ABCD\) are equal, but they are not congruent because the \(2\) figures have equal area, which doesn't mean they have equal sides.
Now, consider the below planar region.
We can see that the planar region \(X\) is made up of \(2\) planar regions \(A\) and \(B\). Therefore, the area of the planar region \(X\) can be written as:
Area of figure \(X =\) Area of figure \(A +\) Area of figure \(B\)
We can denote the area of figure \(A\) as \(ar(A)\) and area of figure \(B\) as \(ar(B)\), and so on.
Now, we arrive at \(2\) properties associated with the planar region. They are:
1. If \(A\) and \(B\) are two congruent figure, then \(ar(A) = ar(B)\).
2. If a planar region \(X\) is formed by \(2\) non-overlapping planar regions \(A\) and \(B\), then \(ar(T) = ar(A) + ar(B)\).
In this chapter, let us discuss the areas of certain geometrical figures under the condition when they lie on the same base and between the same parallels.