ICSE Class 10 Maths Chapter 16 Similarity As a Size Transformation

Chapter 16

Similarity (As a Size-Transformation)

Points To Remember

1. Similarity of Figures

Any two figures are said to be similar, if they have exactly the same shape but not necessarily the same size.

Example:

(i) Two equilateral triangles are always similar.

(ii) Two squares are always similar.

(iii) Two circles are always similar.

Similar Figures

2. Similarity As a Size Transformation

(i) Enlargement: Let us be given a figure, say a quadrilateral ABCD. With the help of this quadrilateral, we shall construct a similar quadrilateral, each of whose sides is twice the corresponding sides of ABCD.

Method: Mark a point P outside ABCD. Join PA, PB, PC and PD.

Produce them to A', B', C' and D' respectively such that:

PA' = 2PA, PB' = 2PB, PC' = 2PC and PD' = 2PD.

Join A' B', B'C', C'D' and D'A'.

Then, A'B'C'D' is called the image of ABCD.

On measurement, it will be found that:

A'B' = 2AB, B'C' = 2BC, C'D' = 2CD and D'A' = 2DA.

We say that the object ABCD has been enlarged by a scale factor 2 about the centre of enlargement P to give the image A'B'C'D'.

Teacher's Note

Enlargement and reduction transformations are used in photocopiers and digital photography to create different sized versions of images while maintaining their shape.

3. Reduction

(ii) Reduction: Let us be given a figure, say a triangle ABC. With the help of this triangle, we shall construct a similar triangle, each of whose sides is equal to half the corresponding sides of triangle ABC.

Method: Take a point P outside triangle ABC.

Join PA, PB and PC.

Mark points A', B' and C' on these line segments such that:

PA' = (1/2)PA, PB' = (1/2)PB and PC' = (1/2)PC.

Join A'B', B'C' and C'A'. Then, triangle A'B'C' is the image of triangle ABC. On measurement, it will be found that A'B' = (1/2)AB, B'C' = (1/2)BC and C'A' = (1/2)CA.

Thus, triangle ABC has been reduced by a scale factor \(\frac{1}{2}\) about the centre of reduction P to give the image A' B' C'.

3. Size Transformation

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

The given figure is called an object or the pre-image and the resulting figure is called its image.

4. Properties of Size Transformation

(i) In a size transformation, the shape of the given figure is preserved.

Thus, angle, perpendicularity, parallelism etc. are preserved.

(ii) Let k be the scale factor of a given size transformation. Then,

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

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

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

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

(iv) Area of the resulting figure = k² (Area of the given figure).

(v) In case of solids, we have

Volume of the resulting figure = k³ (Volume of the given figure).

Teacher's Note

Understanding scale factors helps in map reading and architectural blueprints, where measurements need to be proportionally adjusted to fit practical requirements.

5. 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 Factor, \(k = \frac{1}{p}\).

(i) Length of the model = k \(\times\) (Length of the actual figure).

(ii) Area of the model = k² \(\times\) (Area of the actual figure).

(iii) Volume of the model = k³ \(\times\) (Volume of the actual figure).

6. Map

Let the map of a plane figure be drawn to the scale 1 : p.

Then, Scale Factor, \(k = \frac{1}{p}\).

(i) Length in the map = k \(\times\) (Actual length).

(ii) Area in the map = k² \(\times\) (Actual area).

Teacher's Note

Maps use scale factors to represent large geographical areas on manageable pieces of paper, making travel planning and navigation possible in daily life.

Exercise 16

Q.1. Triangle ABC with sides AB = 3.6 cm, BC = 4.5 cm and CA = 6 cm is enlarged to triangle A'B'C' such that the largest side of the enlarged triangle is 10 cm. Find the scale factor and use it to find the lengths of the other sides of triangle A'B'C'.

Sol. In triangle ABC, AB = 3.6 cm, BC = 4.5 cm and CA = 6 cm and it has been enlarged to triangle A'B'C'. In the resulting triangle A'B'C', largest side = 10 cm.

But largest side in the given triangle ABC = CA = 6 cm.

Scale factor = \(\frac{10}{6} = \frac{5}{3}\).

\(\therefore\) A'B' = 3.6 \(\times\) \(\frac{5}{3}\) = 6.0 cm.

B'C' = 4.5 \(\times\) \(\frac{5}{3}\) = 7.5 cm Ans.

Q.2. A triangle ABC with sides AB = 16 cm, BC = 12 cm and CA = 18 cm is reduced to triangle A'B'C' such that the smallest side of the image triangle is 4.8 cm. Find the scale factor and use it to find the lengths of the other sides of triangle A'B'C'.

Sol. In triangle ABC, AB = 16 cm, BC = 12 cm and CA = 18 cm and it has been reduced to triangle A'B'C'

In the resulting triangle A'B'C', smallest side = 4.8 cm.

But, the smallest side in triangle ABC, BC = 12 cm.

Scale factor = \(\frac{4.8}{12} = \frac{48}{120} = \frac{2}{5}\)

\(\therefore\) A'B' = 16 \(\times\) \(\frac{2}{5}\) = \(\frac{32}{5}\) = 6.4 cm.

C'A' = 18 \(\times\) \(\frac{2}{5}\) = \(\frac{36}{5}\) = 7.2 cm. Ans.

Q.3. A triangle PQR is reduced by a scale factor 0.72. If the area of triangle PQR is 62.5 cm², find the area of its image.

Sol. Scale factor = 0.72

And area of triangle PQR = 62.5 cm²

\(\therefore\) Area of its image triangle P'Q'R'

= 62.5 \(\times\) (0.72)² cm²

= 62.5 \(\times\) 0.72 \(\times\) 0.72 cm²

= 32.4 cm² Ans.

Q.4. A rectangle having an area of 60 cm² is transformed under enlargement about a point in space. If the area of its image is 135 cm², find the scale factor of the enlargement.

Sol. Area of the given rectangle = 60 cm²

And area of its enlargement = 135 cm²

Let, scale factor = k

\(\therefore\) 135 = 60 \(\times\) k² (\(\therefore\) In area)

\(\Rightarrow\) k² = \(\frac{135}{60} = \frac{9}{4}\) \(\Rightarrow\) k = \(\sqrt{\frac{9}{4}} = \frac{3}{2}\)

\(\therefore\) k = 1.5

Hence, scale factor of enlargement = 1.5 Ans.

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