ICSE Class 9 Maths Chapter 12 Similarity

12 Similarity

Points To Remember

1. Similar figures - Two figures are similar if they have the same shape. Similar figures may differ in size. The sign - is used for similarity.

2. Similar Triangles - Two triangles ABC and DEF are said to be similar, if their corresponding sides are proportional and we write, \(\triangle ABC \sim \triangle DEF\).

Thus, \(\triangle ABC \sim \triangle DEF\)

\[\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}\]

3. Axioms Of Similarity Of Triangles

(i) AA - axiom of Similarity - If two triangles have two pairs of corresponding angles equal, then the triangles are similar.

In the given figure, \(\triangle ABC\) and \(\triangle DEF\) are such that \(\angle A = \angle D\) and \(\angle B = \angle E\). Therefore, \(\triangle ABC \sim \triangle DEF\).

(ii) SAS - axiom of Similarity - If two triangles have a pair of corresponding angles equal and the sides including them proportional, then the triangles are similar.

In the given figure, \(\triangle ABC\) and \(\triangle DEF\) are such that \(\angle A = \angle D\) and \(\frac{AB}{DE} = \frac{AC}{DF}\). Therefore, \(\triangle ABC \sim \triangle DEF\).

(iii) SSS - axiom of Similarity - If two triangles have three pairs of corresponding sides proportional then the triangles are similar.

In the given figure, \(\triangle ABC\) and \(\triangle DEF\) are such that \(\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}\). Therefore, \(\triangle ABC \sim \triangle DEF\).

Teacher's Note

Similar triangles appear everywhere in real life - from the shadows cast by objects to the proportions used in maps and architectural blueprints. Understanding similarity helps us calculate heights of buildings or distances using simpler scaled measurements.

4. Size Transformation - It is the process in which a given figure is enlarged or reduced by the scale factor k such that the resulting figure is similar to the given figure.

The given figure is called an object on the Preimage and the resulting figure is called its image.

5. Properties Of Size-Transformation

(i) In the size transformation, the shape of the figure is preserved. Thus angle, perpendicular ship, parallelism etc. are preserved.

(ii) Let k be the scale factor of the given size transformation, then

(a) \(k > 1 \Rightarrow\) The transformation is enlargement.

(b) \(k < 1 \Rightarrow\) The transformation is reduction.

(c) \(k = 1 \Rightarrow\) The transformation is an identity transformation.

(iii) Each side of resulting image = k times the corresponding side of the given object.

(iv) Area of resulting image = \(k^2 \times\) (Area of given object).

(v) Volume of resulting image = \(k^3 \times\) (Volume of given object).

6. Model - The model of a plane figure and the actual figure are similar to one another. Let the model of a plane figure be drawn to the scale 1 : p, then

\[scale \text{ } factor, k = \frac{1}{p}\]

(a) Length of model = k x (Length of actual object).

(b) Area of model = \(k^2 \times\) (Area of actual object).

(c) Volume of model = \(k^3 \times\) (Volume of actual object).

7. Map - Let the map of a plane figure be drawn to the scale 1 : p, then

\[scale \text{ } factor, k = \frac{1}{p}\]

(a) Length in the map = k x (Actual length).

(b) Area in the map = \(k^2 \times\) (Actual area).

8. Theorems - (i) (Basic Proportionality Theorem) A line drawn parallel to one side of a triangle divides the other two sides proportionally.

(ii) (Converse of Theorem) If a line divides any two sides of a triangle proportionally, the line is parallel to the third side.

Teacher's Note

The Basic Proportionality Theorem is fundamental in surveying and construction - when architects draw parallel lines in building designs or engineers create scaled sections of infrastructure, they rely on the principle that proportional division maintains geometric relationships.

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