• Two figures are said to be on same base and between the same parallels, if they have a common side (base) and the vertices (or the vertex) opposite to the common base of each figure lie on a line parallel to the base.
• Figures on the same base and between the same parallels :
In fig.(i) triangles ABC and DBC have a common side, BC. So we can say that Δ’s ABC and DBC are on the same base BC. Similarly, in figs.(ii), (iii), (iv) this condition can be checked.
Area Axioms: The concept of area possesses properties similar to the concept of the length of a line segment. So, it is natural to think that there may be some analogy between the two concepts:
(a) The concept of length
(b) The concept of area.
Area of a figure is a number associated with the part of the plane enclosed by the figure .
• Two congruent figures have equal area but the converse need not to be true.
• A diagonal of a || gm divides it into two triangles of equal area.
ar (ΔABD) = ar (ΔCDB)
• The area of a || gm is the product of its base and corresponding altitude,
ar (|| gm ABCD)
= Base × Altitude
• Parallelograms on the same base and between the same parallels are equal in area.
• Triangles on the same base and between the same parallels are equal in area.
• The area of a triangle is half the product of any of its sides and the corresponding altitude.
• In given fig., ar(△ABC) = ½ (BC × AL)
• The area of a trapezium is half the product of its height and the sum of parallel sides.
• The area of a rhombus is half the product of the length of its diagonals.
• Triangles having equal areas and having one side of the triangle, equal to one side of the other, have their corresponding altitudes equal.
Diagonals of a parallelogram divide it into four triangles of equal area.
• If the diagonals AC and BD of quad. ABCD intersect at O and separate the quad. Into four triangles of equal area, then the quad. ABCD is a parallelogram.
• A median of a triangle divides it into two triangles of equal area.