Similarity Class 10 ICSE — A Das Gupta Chapter 16
Similarity class 10 ICSE, in A. Das Gupta’s Mathematics textbook Chapter 16, explains how figures that share a shape can be enlarged or reduced by a size transformation. The chapter defines similar figures, shows how to build them from a centre point, and connects the scale factor to lengths, areas and volumes. It then applies the same ideas to maps and models drawn to scale.
ICSE Class 10 Mathematics A. Das Gupta Chapter 16 Similarity As a Size Transformation PDF
What is inside Chapter 16
5 pages · 3 sections · 1 worked examples · 10 exercise questions
| Section | Title |
|---|---|
| 1 | Similarity of Figures |
| 2 | Similarity As a Size Transformation |
| 6 | Map |
- EXERCISE 16: 10 questions
1 Similarity of Figures
Two figures are similar when they have exactly the same shape, but not necessarily the same size. Their matching angles are equal, and their matching sides are in proportion. This single idea controls everything else in the chapter.
The chapter opens with three standard examples that are always similar:
- Two equilateral triangles are always similar.
- Two squares are always similar.
- Two circles are always similar.
The reason sits inside each shape. Every equilateral triangle has three equal angles of 60^\circ, so any two match through one scale factor. Every square has four right angles and equal sides, so one square is a scaled copy of another. A circle is fixed by its radius alone, so all circles share the same shape.
This definition anchors the chapter. Once two figures are similar, every rule that follows — for sides, areas and volumes — depends on the scale factor. You can compare this construction view with the earlier chapter on similarity.
Students preparing for the board can confirm the Class 10 Mathematics syllabus on the official CISCE website.
2 Similarity As a Size Transformation
Similar figures can be produced from a given figure by a construction through a fixed point. The construction is called a size transformation, and the fixed point is the centre of enlargement or reduction.
Enlargement. Take a quadrilateral ABCD and a point P outside it. The steps are:
- Join P to each vertex to get PA, PB, PC and PD.
- Produce each segment beyond the vertex so that PA' = 2PA, PB' = 2PB, PC' = 2PC and PD' = 2PD.
- Join the image points A'B'C'D'.
Measuring the image shows A'B' = 2AB, B'C' = 2BC, C'D' = 2CD and D'A' = 2DA. ABCD has been enlarged by a scale factor 2 about the centre of enlargement P to give the image A'B'C'D'.
Worked pattern. If AB is 3 cm, the enlarged side is A'B' = 2 \times 3 = 6 cm. Every other side follows the same pattern.
Reduction. Now take a triangle ABC and a point P outside it:
- Join PA, PB and PC.
- Mark A’, B’ and C’ on these segments so that PA' = \frac{1}{2}PA, PB' = \frac{1}{2}PB and PC' = \frac{1}{2}PC.
- Join A’B’, B’C’ and C’A’.
Measurement gives A'B' = \frac{1}{2}AB, B'C' = \frac{1}{2}BC and C'A' = \frac{1}{2}CA. The triangle has been reduced by a scale factor \frac{1}{2} about the centre of reduction P.
The two constructions differ in one detail. In enlargement, the image points lie beyond the object on produced segments. In reduction, they lie on the original segments, between P and the object. Both preserve the shape, so the image is similar to the object.
3 Size Transformation
A size transformation is the process in which a given figure is enlarged or reduced by a scale factor k, so that the resulting figure is similar to the given figure.
Two names matter here. The given figure is the object, also called the pre-image. The resulting figure is its image. The word “pre-image” simply marks the figure that exists before the transformation.
The name of the process says what changes: the size. The shape stays the same, and every length is multiplied by k. A factor of 2 doubles lengths, and a factor of \frac{1}{2} halves them. The image and the object therefore have equal angles and proportional sides, which is the definition of similar figures from Section 1.
Section 2 showed both cases. The quadrilateral ABCD was the object, and A'B'C'D' was its image after enlargement by 2. The triangle ABC was the object, and \triangle A'B'C' was its image after reduction by \frac{1}{2}. In each construction the image is similar to the object.
4 Properties of Size Transformation
A size transformation preserves the shape of the given figure. Angles, perpendicularity and parallelism are all preserved: a right angle stays a right angle, and parallel sides stay parallel. Only the size changes.
The scale factor k decides the type of transformation:
| Scale factor | Effect on the image | Type |
|---|---|---|
| k \gt 1 | Image is larger than the object | Enlargement |
| k = 1 | Image is the same size as the object | Identity transformation |
| 0 \lt k \lt 1 | Image is smaller than the object | Reduction |
The side rule is direct: each side of the image equals k times the corresponding side of the object. The area rule follows from it. Area is a product of two lengths, and each length is multiplied by k, so the image area equals k^2 times the object area.
Worked pattern 1. A triangle with sides 5 cm, 7 cm and 9 cm is enlarged so the largest image side is 13.5 cm. The largest object side is 9 cm, so k = \frac{13.5}{9} = 1.5. The remaining sides are 5 \times 1.5 = 7.5 cm and 7 \times 1.5 = 10.5 cm.
Worked pattern 2. A square of area 25\ \text{cm}^2 is enlarged with k = 2. Its image area is 25 \times 2^2 = 100\ \text{cm}^2.
Volume misprint note. The book prints property (v) as image volume equals k^2 times object volume. That is a misprint. Volume is a product of three lengths, so the correct factor is k^3. The Model section and every exercise in this chapter use k^3, so use k^3 in your answers.
5 Model
A model of a figure and the actual figure are similar to one another. A model is usually drawn to a scale written as 1:p. The scale means that 1 unit on the model represents p units on the actual figure, so the scale factor is k = \frac{1}{p}.
Because the two figures are similar, three relations hold:
- Length of the model = k \times length of the actual figure.
- Area of the model = k^2 \times area of the actual figure.
- Volume of the model = k^3 \times volume of the actual figure.
The reverse direction is the one most questions need. Since k = \frac{1}{p}, an actual length equals p times the model length. This reverses the first relation.
Worked pattern. A room is modelled at scale 1:40, and the model height is 12 cm. The actual height is 12 \times 40 = 480\ \text{cm} = 480 \div 100 = 4.8\ \text{m}.
6 Map
A map of a plane figure is also drawn to a scale 1:p. The scale factor is therefore k = \frac{1}{p}, exactly as in the model section. Two relations follow:
- Length in the map = k \times actual length.
- Area in the map = k^2 \times actual area.
Map questions usually start from a measurement on the map and ask for the real distance. Use the reverse form: actual length = \frac{1}{k} \times map length. Since k = \frac{1}{p}, this is the same as multiplying the map length by p.
Unit conversion. Map lengths are in centimetres, but answers are expected in kilometres. Use 100\ \text{cm} = 1\ \text{m} and 1000\ \text{m} = 1\ \text{km}, so 100000\ \text{cm} = 1\ \text{km}. The rule is: actual length in km = (map length in cm \times\ p) \div 100000.
Worked pattern. A straight road measures 4 cm on a map at scale 1:50000. The actual length is 4 \times 50000 = 200000\ \text{cm} = 200000 \div 100000 = 2\ \text{km}.
Pythagoras tip. Map questions often join the scale factor with Pythagoras. In this exercise, the triangular plot has two given sides and a right angle between them, so the third side needs a^2 + b^2 = c^2 before the scale is applied. The rectangular plot asks for a diagonal, which is Pythagoras first and the scale factor second. When a map question mentions a right angle or a diagonal, expect both ideas.
Geometry continues in the next chapter on circles.
What to practise from this chapter
This chapter has one exercise block: EXERCISE 16, with 10 questions. The questions group by skill:
- Q1 and Q2 find the scale factor from one given side. Q1 enlarges a triangle; Q2 reduces one.
- Q3 and Q4 apply the area rule k^2. Q4 works backwards, finding k from the two areas.
- Q5 to Q7 join the map scale with Pythagoras and centimetre-to-kilometre conversion.
- Q8 to Q10 use models, including volume at k^3.
Attempt Q5 and Q7 with extra care, because each combines two ideas: the scale factor and Pythagoras. Q10 carries a board-year tag (2009) and tests the volume rule in a past-paper format.
All the Class 10 Mathematics chapters are gathered on the book hub, and the full set of ICSE Class 10 books is listed there too. For other subjects, browse the Class 10 resource hub.
Frequently asked questions
What does the scale factor k tell you about the image?
If k \gt 1, the image is larger than the object, so the transformation is an enlargement. If 0 \lt k \lt 1, the image is smaller, so it is a reduction. If k = 1, the image is the same size and the transformation is an identity transformation.
How do you find the scale factor when only one side of the image is given?
Match the given image side with its corresponding object side, then divide the image length by the object length. Exercise 16 Q1 divides the largest image side by the largest object side. Once you have k, multiply every other object side by it.
How do you convert a map length in centimetres into the actual length in kilometres?
Multiply the map length by the scale denominator p, then divide by 100000. This uses 100\ \text{cm} = 1\ \text{m} and 1000\ \text{m} = 1\ \text{km}. A 4 cm length on a 1:50000 map gives 4 \times 50000 \div 100000 = 2\ \text{km}.
Why is area scaled by k squared when lengths are scaled by k?
Area is a product of two lengths, and each length is multiplied by k, so the area is multiplied by k \times k = k^2. Volume is a product of three lengths, so it is multiplied by k^3. That is why the volume rule in this chapter uses k^3, not k^2.
Reference: ICSE Class 10 Mathematics textbook material, chapter 16: Similarity As a Size Transformation — CISCE-aligned.
More from this book
Explore Class 10 Maths Notes
Related chapters: