If we have a triangular field and want to divide it in to 3 equal parts. Can we divide it with out finding the area. Suppose we divide only one side of the triangular field in three equal parts and joint the points to the opposite vertex. is it equal in size?. We are handling this type of situation in this chapter. to check this we have to find the area of the each plane. The area of the plane is defined as the magnitude of the planar region. And the planar region is the plane enclosed by the simple closed figure. We are also know about the congruent of figures. Two figures are congruent if they have in same shape and same size. then we can keep one figure over other. And if two figure are congruent then they are in same size. But the converse of the statement is not possible. Area of a figure can be write as some number of unit ex 12 cm sq.
In the figure 9.3-p-153. The figure T formed by two planar region formed by figure P and Q. wecan also denote the area of the figure A as ar(A) . The properties of the area of the figure is as follows
- If A and B are two congruent figures then ar(A) = ar(B)
- If a planar region is formed by a figure T is made up of two non-overlapping planar regions formed by figure P and Q then ar(T) = ar(P) + ar(Q).
Figures on same base and between same parallel
See figure 9.4 and 9.5 on p-154.
Figures are said to be on the same base and between the same parallels, if they have common base (side) and the vertices (or vertex)opposite to the common base of each figure lies on a line parallel to the base.
see figure 9.6 and 9.7 on p-155 see exercise also.
parallelograms on the same base and between the same parallels
please see the activity 1and activity 2 in page 156/157. From this we can understand that, the parallelogram on the same base and between the same parallel are equal in area.
Theorem : 9.1 – parallelogram on the same base and between the same parallels are equal in area.
Proof: See page number 157 for details . See page 158,159 for example and exercise.