Class 09 To verify that the ratio of the areas of a parallelogram and a triangle on the same base and between the same parallels is 2:1.

 Activity 21

OBJECTIVE                                                                    

To verify that the ratio of the areas of a parallelogram and a triangle on the same base and between the same parallels is 2:1.

METHOD OF CONSTRUCTION

 MATERIAL REQUIRED

Plywood sheet of convenient size, graph paper, colour box, a pair of wooden strips, scissors, cutter, adhesive, geometry box.

 1.   Take a rectangular plywood sheet.

 2.   Paste a graph paper on it.

 3.   Take any pair of wooden strips or wooden scale and fix these two horizontally so that they are parallel.

 4.   Fix any two points A and B on the base strip (say Strip I) and take any two points C and D on the second strip (say Strip II) such that AB = CD.

 Take any point P on the second strip and join it to A and B [see Fig. 1].

DEMONSTRATION

 1.   AB is parallel to CD and P is any point on CD.

 2.   Triangle PAB and parallelogram ABCD are on the same base AB and between the same parallels.

 3.   Count the number of squares contained in each of the above triangle and

 1parallelograms, keeping half square as ₂ and more than half as 1 square, leaving those squares which contain less than half square.

 4. See that area of the triangle PAB is half of the area of parallelograms ABCD.

 OBSERVATION

 1.   The number of squares in triangle PAB =...............

 2.   The number of squares in parallelogram ABCD =............... .

 So, the area of parallelogram ABCD = 2 [Area of triangle PAB] Thus, area of parallelogram ABCD : area of DPAB = ........ : ...........

APPLICATION

 This activity is useful in deriving formula for the area of a triangle and also in solving problems on mensuration.

Note

 You may take different triangles PAB by taking different positions of point P and the two parallel strips as shown in Fig. 2.

Class 09 To verify the algebraic identity : a3 + b3 = (a + b) (a2 – ab + b2)

 Activity 9

 OBJECTIVE     

                                                             

To verify the algebraic identity : a³ + b³ = (a + b) (a² – ab + b²)

  MATERIAL REQUIRED

Acrylic sheet, glazed papers, saw, adhesive, cellotape, coloured papers, sketch pen, etc.

 METHOD OF CONSTRUCTION

 1.   Make a cube of side a units and another cube of side b units as shown in Fig. 1 and Fig. 2 by using acrylic sheet and cellotape/adhesive.

 2.   Make a cuboid of dimensions a × a × b [see Fig. 3].

 3.   Make a cuboid of dimensions a × b × b [see Fig. 4].

 Arrange these cubes and cuboids as shown in Fig.

DEMONSTRATION

 Volume of cube in Fig. 1 = a³

 Volume of cube in Fig. 2 = b³

 Volume of cuboid in Fig. 3 = a²b

 Volume of cuboid in Fig. 4 = ab²

 Volume of solid in Fig. 5 = a³+b³ + a²b + ab² = (a+b) (a² + b²)

 Removing cuboids of volumes a²b and ab², i.e.,Fig. 6

ab (a + b) from solid obtained in Fig. 5, we get the solid in Fig. 6.

 Volume of solid in Fig. 6 = a³ + b³.

 Therefore,         a³ + b³ = (a+b) (a² + b²) – ab (a + b)

 = (a+b) (a² + b² – ab)

 Here, volumes are in cubic units.

 OBSERVATION

 On actual measurement:

 a = ..............,          b = ..............,

 So, a³ = ..............,       b³  = .............., (a+b) = ..............,    (a+b)a² = ..............,

 (a+b) b²  = ..............,       a²b = ..............,             ab² = ..............,

 ab (a+b)  = ..............,

 Therefore, a³ + b³ = (a + b) (a² + b² – ab).

 APPLICATION

 The identity may be used in simplification and factorisation of algebraic expressions. 

Class 09 To verify that the triangles on the same base and between the same parallels are equal in area.

 

Activity 20

 OBJECTIVE                                                                 

To verify that the triangles on the same base and between the same parallels are equal in area.

 MATERIAL REQUIRED

A piece of plywood, graph paper, pair of wooden strips, colour box , scissors, cutter, adhesive, geometry ox.

METHOD OF CONSTRUCTION

1.   Cut a rectangular plywood of a convenient size.

 2.   Paste a graph paper on it.

 3.   Fix any two horizontal wooden strips on it which are parallel to each other.

4.   Fix two points A and B on the paper along the first strip (base strip).

5.   Fix a pin at a point, say at C, on the second strip.

6.   Join C to A and B as shown in Fig. 1.

7.   Take any other two points on the second strip say C^′ and C^′′ [see Fig. 2].

 8.   Join C^′A, C^′B, C^′′A and C^′′B to form two more triangles.

DEMONSTRATION

 1. Count the number of squares contained in each of the above triangles, taking

 1 half square as ₂ and more than half as 1 square, leaving those squares which1

contain less than       squares.

 2.    See that the area of all these triangles is the same. This shows that triangles on the same base and between the same parallels are equal in area.

 OBSERVATION

 1.   The number of squares in triangle ABC =.........., Area of ΔABC = ........ units

 2.   The number of squares in triangle ABC^′ =......., Area of D ABC^′ = ........ units

 3.   The number of squares in triangle ABC^′′ =....... , Area of D ABC^′′ = ........ units Therefore, area (ΔABC) = ar(ABC^′) = ar(ABC^′′).

 APPLICATION

 This result helps in solving various geometric problems. It also helps in finding the formula for area of a triangle. 

Fig. 1

 

Class 09 To verify experimentally that the parallelograms on the same base and between same parallels are equal in area.

 

Activity 19

 OBJECTIVE                                                                    

To verify experimentally that the parallelograms on the same base and between same parallels are equal in area.

 MATERIAL REQUIRED

 A piece of plywood, two wooden strips, nails, elastic strings, graph paper.

 METHOD OF CONSTRUCTION

 1.   Take a rectangular piece of plywood of convenient size and paste a graph paper on it.

 2.   Fix two horizontal wooden strips on it parallel to each other [see Fig. 1].

Fig. 1

 3.   Fix two nails A₁ and A₂ on one of the strips [see Fig. 1].

 4.   Fix nails at equal distances on the other strip as shown in the figure.

 DEMONSTRATION

 1.   Put a string along A₁, A₂, B₈, B₂ which forms a parallelogram A₁A₂B₈B₂. By counting number of squares, find the area of this parallelogram.

2.   Keeping same base A₁A₂, make another parallelogram A₁A₂B₉B₃ and find the area of this parallelogram by counting the squares. 

3.   Area of parallelogram in Step 1 = Area of parallelogram in Step 2.

OBSERVATION

Number of squares in 1st parallelogram = --------------.

 Number of squares in 2nd parallelogram = -------------------.

 Number of squares in 1st parallelogram = Number of squares in 2nd parallelogram.

 Area of 1st parallelogram = --------- of 2nd parallelogram

APPLICATION

This result helps in solving various geometrical problems. It also helps in deriving the formula for the area of a paralleogram.