Congruence And Similarity
In this chapter, we will be discussing the belowmentioned topics in detail:
 Scale Factor
 Enlargement and Reduction of Plane Figures
 Map and Scales
 Length Scale
CONGRUENCE 
SIMILARITY 

Shape 
Same 
Same 
Size 
Same 
May be Different 
SCALE FACTOR
Scale Factor comes into the picture when we either reduce i.e. make an image smaller or enlarge the original image i.e. make it bigger.
Let’s understand this with the help of some examples:
Question 1:
\(∆PQR\) is formed by scaling \(∆ABC\). What is the scale factor?
Original
Resultant
Solution:
\(∆ABC\) is the original triangle and \(∆PQR\) is the resultant triangle.
\(\begin{align*} Scale\;factor &= \frac{Length \;Of \;Side \;In \;Resultant \;Image}{Length \;Of \;Corresponding \;Side \;In \;Original \;Image} \\ \\ \frac{PQ}{AB} &= \frac{5\;cm}{3\;cm} \\ \\ &= 1\frac23 \end{align*}\)
Question 2:
\(∆ABC\) is similar to \(∆PQR\). \(∆PQR\) is formed by scaling \(∆ABC\). What is the scale factor?
What can we say about the relationship between the scale factor and the resultant image?
Solution:
\(∆ABC\) is the original triangle and \(∆PQR\) is the resultant triangle.
Scale factor \(\begin{align*} &= \frac{Length \;of \;side \;in \;Resultant \;Image}{Length \;of \;Corresponding \;side \;in \;original \;Image} \end{align*}\)
\(\begin{align*} \frac{PQ}{AB} &= \frac{4\;cm}{8\;cm}\\ &= \frac12 \end{align*}\)
Hence, the scale factor is less than \(1\) and the resultant image is smaller than the original.
Question 3:
\(∆ABC\) is similar to \(∆PQR\). \(∆PQR\) is formed by scaling \(∆ABC\). What is the scale factor?
What can we say about the relationship between the scale factor and the resultant image?
Solution: \(∆ABC\) is the original triangle and \(∆PQR\) is the resultant triangle.
Scale factor \(\begin{align*} &= \frac{Length \;of \;side \;in \;Resultant \;Image}{Length \;of \;Corresponding \;side \;in \;original \;Image} \end{align*}\)
\(\begin{align*} \frac{PQ}{AB} &= \frac{4\;cm}{4\;cm}\\ &= 1 \end{align*}\)
Hence, the scale factor is equal to \(1\) and the resultant image is of the same size as the original.
Scale Factor
In general, a figure and its resultant image are similar.
Scale Factor 
Resultant Image 
\(k > 1\) 
It would be enlarged i.e. resultant image would be bigger than the original. 
\(0 < k < 1\) 
Reduced/ Smaller than the original 
\(k = 1\) 
It would be unchanged i.e. congruent 
Alternate Form of Scale Factor Formula
Question 4:
\(∆ADE\) is an enlargement of \(∆ABC\) with a scale factor of \(\begin{align*} 1\frac {4}{9} \end{align*}\). Given that \(AC = 27cm\), find the length of \(CE\).
Solution:
Method 1:
\(∆ADE\) and \(∆ABC\) are similar because it is just an enlargement.
\(\begin{align*} Scale \;factor&= \frac{Length \;of \;side \;in \;Resultant \;Image}{Length \;of \;Corresponding \;side \;in \;original \;Image} \\ \\ \frac{AE}{AC} &= 1\frac49\\ \\ \frac{AC+CE}{AC} &= \frac{13}9\\ \\ \frac{27+CE}{27} &= \frac{13}9\\ \\ CE &= 27 \bigg(\frac{13}9\bigg)  27\\ \\ &= 12 \;cm \end{align*}\)
Method 2:
\(\begin{align*} Resultant \;Length &= Scale \;Factor × Original \;Length \\ \\ AE &= 1\frac49 × AC\\ \\ &= 1\frac49 × 27\\ \\ &= 39 \;cm\\ \\ CE &= AE \;–\;AC\\ \\ &= 39 \;– \;27\\ \\ &= 12 \;cm \end{align*}\)
Concept of Ratio
A ratio compares two quantities of the same kind that either has no units or are measured in the same units.
The ratio \(a : b\), where \(a\) and \(b\) are positive integers, has no units.
Length Scale
For example, a map has a scale where \(1 \;cm\) represents \(1 \;km\). Express the map scale as a ratio in the form \(1: n\), where \(n\) is an integer.
\(\begin{align*} \\1\;cm &: 1\; km\\ \\ 1\;cm &: 1000 \;m\\ \\ 1\;cm &: 100000 \;cm\\ \\ 1 &: 100000 \end{align*}\)
Let’s understand this with the help of some examples:
Question 5:
A map has a scale of \(1 \;cm\) to \(4 \;km\). Find
 the actual distance, in \(km\), is represented by \(6.8 \;cm\) on the map.
 the length on the map represents an actual distance of \(15.8 \;km\).
 the scale of the map in the form \(\frac1n\), where \(n\) is an integer.
Solution:
i) Map : Actual
\(1 \;cm : 4 \;km\)
Multiplying both sides of the ratio by \(6.8\),
\(\begin{align*} \\(1 × 6.8) \;cm &: (4 × 6.8) \;km\\ \\ 6.8 \;cm &: 27.2 \;km \end{align*}\)
Hence, the actual distance is \(27.2 \;km\).
ii) Map : Actual
\(1 \;cm : 4 \;km\)
Dividing both sides of the ratio by \(4\),
\(0.25 \;cm : 1 \;km\)
Multiplying by \(15.8\),
\(3.95 \;cm : 15.8 \;km\)
Hence, the length on the map is \(3.95 \;cm\).
iii) \(\begin{align*} 1 \;cm : 4 \;km \end{align*}\)
\(\begin{align*} 1 \;cm &: 4000 \;m\\ 1 \;cm &: 400000 \;cm \end{align*}\)
The scale of the map is \(\frac{1}{400000}\).
Continue Learning  

Algebraic Fractions  Direct & Inverse Proportion 
Congruence And Similarity  Factorising Quadratic Expressions 
Further Expansion And Factorisation  Quadratic Equations And Graphs 
Simultaneous Equation 