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.

Class 09 To verify experimentally that in a triangle, the longer side has the greater angle opposite to it.

 

Activity 18

 OBJECTIVE                                                                    

To verify experimentally that in a triangle, the longer side has the greater angle opposite to it.

MATERIAL REQUIRED

 Coloured paper, scissors, tracing paper, geometry box, cardboard sheet, sketch pens.

 METHOD OF CONSTRUCTION

 1.    Take a piece of cardboard of a convenient size and paste a white paper on it.

 2.   Cut out a ΔABC from a coloured paper and paste it on the cardboard [see Fig. 1].

 3.   Measure the lengths of the sides of ΔABC.

 4.   Colour all the angles of the triangle ABC as shown in Fig. 2.

 5.   Make the cut-out of the angle opposite to the longest side using a tracing paper [see Fig. 3].

DEMONSTRATION

 Take the cut-out angle and compare it with other two angles as shown in Fig. 4.

 ∠A is greater than both ∠B and ∠C.

 i.e., the angle opposite the longer side is greater than the angle opposite the other side.

 OBSERVATION

 Length of side AB = .......................

 Length of side BC = .......................

 Length of side CA = .......................

 Measure of the angle opposite to longest side = .......................

 Measure of the other two angles = ...................... and .......................

 The angle opposite the ...................... side is ...................... than either of the other

 two angles.

 APPLICATION

 The result may be used in solving different geometrical problems.

Class 09 To verify experimentally that the sum of the angles of a quadrilateral is 360º.

 Activity 17

  OBJECTIVE                                                                  

To verify experimentally that the sum of the angles of a quadrilateral is 360º.

  METHOD OF CONSTRUCTION

   MATERIAL REQUIRED

 

Cardboard, white paper, coloured drawing sheet, cutter, adhesive, geometry box, sketch pens, tracing paper.

 1.   Take a rectangular cardboard piece of a convenient size and paste a white paper on it.

 2.   Cut out a quadrilateral ABCD from a drawing sheet and paste it on the cardboard [see Fig. 1].

 Make cut-outs of all the four angles of the quadrilateral with the help of a tracing paper [see Fig. 2]

4. Arrange the four cut-out angles at a point O as shown in Fig. 3.

 DEMONSTRATION

 1.   The vertex of each cut-out angle coincides at the point O.

2. Such arrangement of cut-outs

 shows that the sum of the angles

 of a quadrilateral forms a

 complete angle and hence is

equal to 360º.

 

OBSERVATION				Fig. 3
Measure of	∠A =	----------.
Measure of	∠B =	----------.	Measure of  ∠C =	----------.
Measure of	∠D =	----------.	Sum [ ∠A+ ∠B+ ∠C+ ∠D] =		-------------.

 APPLICATION

 This property can be used in solving problems relating to special types of quadrilaterals, such as trapeziums, parallelograms, rhombuses, etc.

Class 09 To verify exterior angle property of a triangle.

 Activity 16

 OBJECTIVE                                                                   

To verify exterior angle property of a triangle.

 METHOD OF CONSTRUCTION

  MATERIAL REQUIRED

Hardboard sheet, adhesive, glazed papers, sketch pens/pencils, drawing sheet, geometry box, tracing paper, cutter, etc.

1.   Take a hardboard sheet of a convenient size and paste a white paper on it.

 2.   Cut out a triangle from a drawing sheet/glazed paper and name it as ΔABC and paste it on the hardboard, as shown in Fig. 1.

 Produce the side BC of the triangle to a point D as shown in Fig. 2.

 4.   Cut out the angles from the drawing sheet equal to ∠A and ∠B using a tracing paper [see Fig. 3].

 5.   Arrange the two cutout angles as shown in Fig. 4.

 Fig. 3

 Fig. 4

 DEMONSTRATION

 ∠ACD is an exterior angle.

 ∠A and ∠B are its two interior opposite angles.

 ∠A and ∠B in Fig. 4 are adjacent angles.

 From the Fig. 4, ∠ACD = ∠A + ∠B.

 OBSERVATION

 Measure    of    ∠A=  __________,  Measure  of  ∠B  =  __________,

Sum (∠A + ∠B) = ________, Measure of ∠ACD = _______.

 Therefore, ∠ACD = ∠A + ∠B.

 APPLICATION

 This property is useful in solving many geometrical problems.