verify that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.

   

Topic: Ratio of the areas of similar triangles

Objective: 

To verify that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.

Pre-requisite knowledge :

(1) To divide a line segment into given number of equal parts.

(ii) Construction of lines parallel to a given line.

Materials required :

(1) Poster paper

(ii) Paint box

(iii) Ruler

(iv) Pencil/ball point pen.

To perform the activity :

Steps:

1. Draw any triangle ABC (see fig. 2.1) on a poster paper.

2. Divide AB into 5 equal parts at points P₁, P₂, P, and P, and AC also into 5 equal parts at the points Q₁, Q₂, Q3 and Q₁. Then the lines P,Q,, P.Q P,Q, and P,Q, are parallel to the base BC of TriangleABC.

3. Through points P₁, P₂, P3 and P, draw lines parallel to AC and through points Q₁, Q₂ Q3 and Q, draw lines parallel to AB. Thus AABC is divided into smaller Paint these smaller tirangles as shown in fig. 2.1.

4. Draw a triangle DEF (see fig. 2.2) of sides DE=AB, EF =3/5AC, so that triangle DEF & ABC are similar. 

5. Divide DE into 3 equal parts at the points R₁, R₂ and DF also into 3 equal parts at the points S₁, S₂. Then the lines R,S₁, R₂S, are parallel to the base EF of ADEF.

6. Through points R₁, R₂ draw lines parallel to DF and through points. S₁, S₂ draw lines parallel to DE. Thus ADEF is divided into smaller triangles. Paint these smaller triangles as shown in fig. 2.2.

Result:

We observe that:

(i) Triangle ABC is divided into 25 smaller triangles, all congruent to each other and of equal area.

(ii) Triangle DEF is divided into 9 smaller triangles, all congruent to each other and of equal area.

(iii) Each small triangle of TriangleABC is congruent to each small triangle of TriangleDEF and these small triangles have equal area.

Area of triangle DEF/Area of triangle ABC =

           area of 9 small congruent triangle/area of 25 congruent triangle =   9/25

Square of base of TriangleDEF/ Square of base of TriangleABC = ( 3/5BC) ²/BC² = ( 3/5) ² = 9/25

 Area of triangle DEF/Area of triangle ABC

       =  Square of base of triangle DEF/Square of base of triangle ABC

Hence the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.