cbse class 9 – SaitechLabs Paint Consultancy Services

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Figures on the same base and between the same parallels
Parallelograms on the same Base and Between the same Parallels
Triangles on the same Base and between the same ParallelsFigures on the same base and between the same parallels

Key Concepts

Area of a figure is a number associated with the part of the plane enclosed by that figure.
Two congruent figures have equal areas (A and B) but the converse need not be true.

If a planar region formed by a figure T is made up of two non-overlapping planar regions formed by figures P and Q, then ar (T) = ar (P) + ar (Q), where ar (X) denotes the area of figure X.

Two figures are said to be on the same base and between the same parallels, if they have a common base (side) and the vertices, (or the vertex) opposite to the common base of each figure lie on a line parallel to the base.

i) ABCD – trapezium, CDEF – parallelogram. Both of them have common base, DC between two parallels DC and AF.

ii) PQRS and MNRS are parallelograms having same base RS and between two parallels RS and PN

iii) Triangles ABC and BDC have same base, BC and are betweent two same parallels AD and BC.

iv) Parallelogram ABCD and triangle CDP have same base, CD and between two same parallels CD and AP.

Area of a parallelogram is the product of its base (b) and the corresponding altitude (h).

Image result for Area of a parallelogram is the product of its base and the corresponding altitude. — Area of parallelogram = b x h

Parallelograms on the same base (or equal bases) and having equal areas lie between the same parallels.

Parallelograms on the same base (or equal bases) and between the same parallels are equal in area. Area of parallelogram ABCD = Area of parallelogram PQCD. They have the common base = CD; they have the common parallels CD and PB.

If a parallelogram (ABCD) and a triangle (ABP) are on the same base and between the same parallels, then area of triangle is half the area of parallelogram.

Area of triangle ABP = 1/2 (area of parallelogram ABCD) = 1/2 (b x h)

Triangles on the same base (or equal bases) and between the same parallels are equal in area.

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

Image result for Area of a triangle is half the product of its base and the corresponding altitude.

Triangles on the same base (or equal bases) and having equal areas lie between the same parallels.

Image result for Triangles on the same base (or equal bases) and having equal areas lie between the same parallels.

A median (AD) of a triangle divides it into two triangles of equal areas.

Image result for A median of a triangle divides it into two triangles of equal areas.

Guided problems based on figures on the Same Base and Between the Same Parallels

Problem-1

Which of the following figures lie on the same base and between the same
parallels. In such a case, write the common base and the two parallels.

Answer:

(i), (iii) and (v) have common base and same parallel lines.

i) Common base: DC, Same parallel lines: CD and AB.

ii) Has common base, but no common parallel lines.

iii) Common base: QR, Common parallel lines: QR and PS.

iv) No common base, but has common parallel lines.

v) Common base: AD, Common parallel lines: AD and BQ.

vi) No common base, No common parallel lines.

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Guided problems based on Parallelograms on the same Base and Between the same Parallels

Problem-1

In the figure given below, ABCD is a parallelogram. AE is perpendicular to DC and CF is perpendicular to AD. If AB is 16 cm, AE is 8 cm and CF is 10 cm, find AD.