Mathematics Lab Activity-09 Class X | Triangle

 Mathematics Lab Activity-09 Class X

Mathematics Laboratory Activities on Similarity of Triangles for class X students with complete observation tables strictly according to the CBSE syllabus.

Chapter - 06 : triangle

Activity - 09

Objective
To establish the criteria for similarity of two triangles.

Material Required

Coloured papers, glue, sketch pen, cutter, geometry box .

Procedure

CASE 1

1. Take a coloured paper/chart paper. Cut out two triangles ABC and PQR with their corresponding angles equal.

Figure 1

Figure 2

2. In the triangles ABC and PQR, ∠A = ∠P; ∠B = ∠Q and ∠C = ∠R.

3. Place the △ABC on △PQR such that vertex A falls on vertex P and side AB falls along side PQ (side AC falls along side PR) as shown in Fig. 2.

4. In Fig. 2, ∠B = ∠Q  or  ∠C = ∠R  

Since corresponding angles are equal, BC || QR, therefore by BPT, 

 

    ...... By invertendo

5. Adding 1 to both side we get 

 

 

    ........ (1)

6. Place the △ABC on △PQR such that vertex B falls on vertex Q, and side BA falls along side QP (side BC falls along side QR) as shown in Fig. 3.

Figure 3

7. In Fig. 3, ∠C = ∠R. 

Since corresponding angles are equal, therefore by BPT AC || PR

   

 

 

By applying invertendo we get

    .......... (2)   

8. From eqn. 1 and eqn. 2

 

△ABC and △PQR are similar to each other and is called the AAA criterian of similarity.

CASE 2

1. Take a colored paper / chart paper, cut out two triangles ABC and PQR with their corresponding sides proportional.

 

Figure 4

2. Place the △ABC on △PQR such that vertex A falls on vertex P and side AB falls along side PQ. Observe that side AC falls along side PR [see Fig. 4].

In figure 4 

  

So by BPT, BC || QR

3. Now in both the triangles we have 

        ∠B = ∠Q  

        ∠C = ∠R

Also ∠A = ∠P

△ABC and △PQR are similar to each other and is called the SSS criterian of similarity.

CASE 3
1. Take a coloured paper/chart paper, cut out two triangles ABC and PQR such that their one pair of sides is proportional and the angles included between the pair of sides are equal.

Figure 5

2. In △ABC and △PQR,

 

3. Place triangle ABC on triangle PQR such that vertex A falls on vertex P and side AB falls along side PQ as shown in Fig. 5.

4. In figure 5
  

∴ by BPT,  BC || QR

∴    ∠B = ∠Q  

      ∠C = ∠R

5. If two angles of any two triangles are equal then third angle are also equal.

∴    ∠A = ∠P

△ABC and △PQR are similar to each other and is called the SAS criterian of similarity.

Observations & calculations

CASE 1:  By actual measurement:

I. In △ABC and △PQR,

∠A =  ∠P = 80^o 

∠B = ∠Q = 60^o 

∠C = ∠R = 40^o

 

 

If corresponding angles of two triangles are EQUAL, the sides are PROPORTIONAL. Hence the triangles are SIMILAR. and is called AAA sinilarity criterian.

CASE 2 : By actual measurement:

1.  In △ABC and △PQR

 

 
 

∠A =  ∠P = 80^o 

∠B = ∠Q = 60^o 

∠C = ∠R = 40^o 

If the corresponding sides of two triangles are PROPORTIONAL, then their corresponding angles are EQUAL. Hence, the triangles are SIMILAR  and is called SSS similarity criterian.

CASE 3 : By actual measurement:

1.  In △ABC and △PQR

 

 

∠A =  ∠P = 80^o 

∠B = ∠Q = 60^o 

∠C = ∠R = 40^o

If two sides of one triangle are PROPORTIONAL to the two sides of other triangle and angles included between them are EQUAL, then the triangles are SIMILAR and is called SAS similarity criterian.

Result

1) When the corresponding angles of two triangles are equal, then their corresponding sides are proportional.

Hence, the two triangles are similar. This is AAA criterion for similarity of triangles.

2) When the corresponding sides of two triangles are proportional, their corresponding angles are equal. 

Hence, the two triangles are similar. This is the SSS criterion for similarity of two triangles

3) When two sides of one triangle are proportional to two sides of another triangle and the angles included between the two pairs of sides are equal, then corresponding angles of two triangles are equal.

Hence, the two triangles are similar. This is the SAS criterion for similarity of two triangles

Applications

This formula is used to find the centroid of a triangle in geometry, vector algebra and 3–dimensional geometry.

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