ICSE Class 10 Maths Chapter 15 Similarity

15. Similarity

(With Applications to Maps and Models)

15.1 Introduction

1. Similarity of Figures: Two figures are said to be similar, if they have the same shape but may differ in size.

Examples:

(i) Two squares of different sizes.

(ii) Two triangles of different sizes.

(a) Same shape means: Angles of one figure are equal to corresponding angles of other figure each to each.

(b) Same size means: Sides of one figure are equal to corresponding sides of the other figure each to each.

2. Congruency of Figures: Two figures are said to be congruent, if they have the same shape and the same size.

Congruent figures are always similar, whereas similar figures are not necessarily congruent.

15.2 Similar Triangles

Two triangles are said to be similar, if their corresponding angles are equal and corresponding sides are proportional (i.e. the ratios between the lengths of corresponding sides are equal).

e.g. Triangles ABC and PQR are similar, if:

(a) their corresponding angles are equal

i.e. \(\angle A = \angle P\), \(\angle B = \angle Q\) and \(\angle C = \angle R\)

(b) their corresponding sides are in proportion

i.e. \(\frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR}\)

Symbolically, we write: \(\triangle ABC \sim \triangle PQR\); where symbol \(\sim\) is read as, "is similar to".

In the same way;

\(\triangle ABC \sim \triangle DEF \Leftrightarrow \begin{cases} \angle A = \angle F, \angle B = \angle E \text{ and } \angle C = \angle D. \\ \text{Also, } \frac{AB}{EF} = \frac{BC}{DE} = \frac{AC}{DF} \end{cases}\)

15.3 Corresponding Sides and Corresponding Angles

1. In similar triangles, the sides opposite to equal angles are said to be the corresponding sides.

e.g. the adjoining figure shows, \(\triangle ABC \sim \triangle PRQ\) in which \(\angle A = \angle P\), \(\angle B = \angle R\) and \(\angle C = \angle Q\).

\(\therefore\) (i) \(\angle A = \angle P\) \(\Rightarrow\) Sides opposite to \(\angle A\) and \(\angle P\) are the corresponding sides. \(\Rightarrow\) Sides BC and QR are the corresponding sides.

(ii) \(\angle B = \angle R\) \(\Rightarrow\) Sides opposite to \(\angle B\) and \(\angle R\) are the corresponding sides \(\Rightarrow\) Sides AC and PQ are the corresponding sides.

Similarly, (iii) \(\angle C = \angle Q\) \(\Rightarrow\) Sides AB and PR are the corresponding sides.

As, the triangles ABC and PRQ are similar; we have:

\(\frac{BC}{QR} = \frac{AC}{PQ} = \frac{AB}{PR}\) [Corresponding sides of similar triangles are proportional]

In the same way, if in \(\triangle ABC\) and \(\triangle PQR\):

\(\angle A = \angle Q\), \(\angle B = \angle R\) and \(\angle C = \angle P\);

the two triangles are similar

i.e. \(\triangle ABC\) is similar to \(\triangle QRP\)

And, \(\frac{\text{side opp. to } \angle A}{\text{side opp. to } \angle Q} = \frac{\text{side opp. to } \angle B}{\text{side opp. to } \angle R} = \frac{\text{side opp. to } \angle C}{\text{side opp. to } \angle P} \Rightarrow \frac{BC}{PR} = \frac{AC}{PQ} = \frac{AB}{QR}\)

2. In similar triangles, the angles opposite to proportional sides are the corresponding angles and so, they are equal.

e.g. in similar triangles ABC and EFD,

if \(\frac{AB}{EF} = \frac{BC}{DF} = \frac{AC}{DE}\), then

(i) ratio \(\frac{AB}{EF} \Rightarrow\) angle opposite to side AB = angle opposite to side EF

i.e. \(\angle C = \angle D\),

(ii) ratio \(\frac{BC}{DF} \Rightarrow\) angle opposite to side BC = angle opposite to side DF

i.e. \(\angle A = \angle E\) and

(iii) ratio \(\frac{AC}{DE} \Rightarrow\) angle opposite to side AC = angle opposite to side DE

i.e. \(\angle B = \angle F\).

Note 1: Please note that in congruent triangles the corresponding sides are equal, whereas in similar triangles, the corresponding sides are in proportion.

e.g. in the adjoining figures; if \(\triangle ABC\) and \(\triangle FED\) are congruent, then BC = DE, AB = EF and AC = DF.

But, if \(\triangle ABC\) and \(\triangle FED\) are similar, then

\(\frac{BC}{DE} = \frac{AB}{EF} = \frac{AC}{DF}\)

Note 2: Triangles, which are similar to the same triangle, are similar to each other also.

15.4 Condition of Similar Triangle

(SAS, AA or AAA and SSS)

1. If one angle of a triangle is equal to any angle of the other triangle and in both the triangles, the sides including the equal angles are in proportion; then the triangles are similar. (SAS postulate)

Example:

If in \(\triangle ABC\) and \(\triangle DEF\),

\(\angle A = \angle D\) and \(\frac{AB}{DE} = \frac{AC}{DF}\)

then by SAS, \(\triangle ABC \sim \triangle DEF\).

Similarly, if \(\angle B = \angle E\) and \(\frac{AB}{DE} = \frac{BC}{EF}\), then also \(\triangle ABC \sim \triangle DEF\) and so on.

2. If two triangles have atleast two pairs of corresponding angles equal; the triangles are similar. (AA or AAA postulate)

Example:

If in \(\triangle ABC\) and \(\triangle DEF\),

\(\angle A = \angle D\) and \(\angle B = \angle E\), \(\triangle ABC \sim \triangle DEF\).

Since, the sum of the angles of a triangle is \(180°\), therefore if two angles of one triangle are equal to two angles of another triangle each to each, their third angles are also equal.

3. If two triangles have their three pairs of corresponding sides proportional, then the triangles are similar. (SSS postulate)

Example:

If in \(\triangle ABC\) and \(\triangle PQR\),

\(\frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR}\)

\(\triangle ABC \sim \triangle PQR\).

Teacher's Note

Understanding similarity helps us recognize that objects of different sizes can have the same proportions - like how a small model of a building has the same shape as the actual building but in reduced size.

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