# Congruence And Similarity

In this chapter, we will be discussing the below-mentioned topics in detail:

• Scale Factor
Enlargement and Reduction of Plane Figures
• Map and Scales
Length Scale

## Congruence and Similarity

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.

$\small{\bbox[20px, #27aae1, border: 5px solid #262262] { \displaystyle{\text{ }\\[2ex] \color{white}{\text{Scale Factor} = \frac{\text{Length Of Side In Resultant Image}}{\text{Length Of The Corresponding Side In Original Image}}}\\[4ex] } } }$

Let’s understand this with the help of some examples:

Question 1:

$\mathrm{\triangle PQR}$ is formed by scaling $\mathrm{\triangle ABC}$. What is the scale factor?

Original

Resultant

Solution:

$\mathrm{\triangle ABC}$ is the original triangle and $\mathrm{\triangle PQR}$ is the resultant triangle.

\small{\begin{align*} \text{Scale factor} &= \frac{\text{Length Of Side In Resultant Image}}{\text{Length Of Corresponding Side In Original Image}} \\ \\ \mathrm{\frac{PQ}{AB}} &= \mathrm{\frac{5\;cm}{3\;cm}} \\ \\ &= 1\frac{2}{3} \end{align*}}

Question 2:

$\mathrm{\triangle ABC}$ is similar to $\mathrm{\triangle PQR}$. $\mathrm{\triangle PQR}$ is formed by scaling $\mathrm{\triangle ABC}$. What is the scale factor?
What can we say about the relationship between the scale factor and the resultant image?

Solution:

$\mathrm{\triangle ABC}$ is the original triangle and $\mathrm{\triangle PQR}$ is the resultant triangle.

\small{\begin{align*} \text{Sacle Factor} &= \frac{\text{Length Of Side In Resultant Image}}{\text{Length Of Corresponding Side In Original Image}} \\[2ex] \frac{\mathrm{PQ}}{\mathrm{AB}} &= \frac{\mathrm{4\;cm}}{\mathrm{8\;cm}}\\[2ex] &= \frac12 \end{align*}}

Hence, the scale factor is less than $1$ and the resultant image is smaller than the original.

Question 3:

$\mathrm{\triangle ABC}$ is similar to $\mathrm{\triangle PQR}$. $\mathrm{\triangle PQR}$ is formed by scaling $\mathrm{\triangle ABC}$. What is the scale factor?
What can we say about the relationship between the scale factor and the resultant image?

Solution:

$\mathrm{\triangle ABC}$ is the original triangle and $\mathrm{\triangle PQR}$ is the resultant triangle.

\small{\begin{align*} \text{Sacle Factor} &= \frac{\text{Length Of Side In Resultant Image}}{\text{Length Of Corresponding Side In Original Image}} \\[2ex] \frac{\mathrm{PQ}}{\mathrm{AB}} &= \frac{\mathrm{4\;cm}}{\mathrm{4\;cm}}\\[2ex] &= 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 It would be enlarged i.e. resultant image would be bigger than the original. Reduced/ Smaller than the original It would be unchanged i.e. congruent

### Alternate form of Scale Factor Formula

$\small{\bbox[20px, #27aae1, border: 5px solid #262262] { \displaystyle{\text{ }\\[2ex] \color{white}{\text{Scale Factor} = \frac{\text{Length Of Side In Resultant Image}}{\text{Length Of The Corresponding Side In Original Image}}}\\[4ex] } } }$

$\small{\bbox[20px, #27aae1, border: 5px solid #262262] { \displaystyle{ \text{ }\\[2ex] \color{white}{ \text{Length Of Side In Resultant Image} =\text{Scale Factor} \times \text{Original Length} }\\[4ex] } } }$

Question 4:

$\mathrm{\triangle ADE}$ is an enlargement of $\mathrm{\triangle ABC}$ with a scale factor of $\displaystyle{1\frac {4}{9}}$. Given that $\text{AC = 27 cm}$, find the length of $\text{CE}$.

Solution:

Method 1:

$\mathrm{\triangle ADE}$ and $\mathrm{\triangle ABC}$ are similar because it is just an enlargement.

\small{\begin{align*} \text{Scale Factor} &= \frac{\text{Length Of Side In Resultant Image}}{\text{Length Of Corresponding Side In Original Image}} \\[2ex] \mathrm{ \frac{AE}{AC}} &= 1\frac{4}{9}\\[2ex] \mathrm{\frac{AC+CE}{AC}} &= \frac{13}{9}\\[2ex] \mathrm{\frac{27+CE}{27}} &= \frac{13}9\\[2ex] \mathrm{CE} &= 27 \bigg(\frac{13}9\bigg) - 27\\ \\ &= \text{12 cm} \end{align*}}

Method 2:

\small{\begin{align*} \text{Resultant Length} &= \text{Scale Factor × Original Length} \\[2ex] \text{AE} &= 1\frac{4}{9} \times \text{AC}\\[2ex] &= 1\frac{4}{9} × 27\\[2ex] &= \text{39 cm}\\ \\ \text{CE} &= \text{AE – AC}\\ \\ &= 39 - 27\\ \\ &= 12 \text{ 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 $\displaystyle{a : b}$, where $a$ and $b$ are positive integers, has no units.

### Length Scale

For example, a map has a scale where $1 \text{ cm}$ represents $1 \text{ km}$. Express the map scale as a ratio in the form $\displaystyle{1: n}$, where $n$ is an integer.

\small{\begin{align*} 1\text{ cm} &: 1\text{ km}\\[2ex] 1\text{ cm} &: 1000 \text{ m}\\[2ex] 1\text{ cm} &: 100000 \text{ cm}\\[2ex] 1 &: 100000 \end{align*}}

Let’s understand this with the help of some examples:

Question 5:

A map has a scale of $1 \text{ cm}$ to $4 \text{ km}$. Find

1. the actual distance, in $\text{km}$, is represented by $6.8 \text{ cm}$ on the map.
2. the length on the map represents an actual distance of $15.8 \text{ km}$.
3. the scale of the map in the form $\displaystyle{\frac{1}{n}}$, where $n$ is an integer.

Solution:

1. Map : Actual

$\small{1 \text{ cm} : 4 \text{ km}}$

Multiplying both sides of the ratio by $\small{6.8}$,

\small{\begin{align*} (1 × 6.8) \text{ cm} &: (4 × 6.8) \text{ km}\\[2ex] 6.8 \text{ cm} &: 27.2 \text{ km} \end{align*}}

Hence, the actual distance is $\small{27.2 \text{ km}}$.

1. Map : Actual

$\small{1 \text{ cm} : 4 \text{ km}}$

Dividing both sides of the ratio by $\small{4}$,

$\small{0.25 \text{ cm} : 1 \text{ km}}$

Multiplying by $\small{15.8}$,

$\small{3.95 \text{ cm} : 15.8 \text{ km}}$

Hence, the length on the map is $\small{3.95 \text{ cm}}$.

1.

\small{\begin{align*} 1 \text{ cm} &: 4 \text{ km}\\[2ex] 1 \text{ cm} &: 4000 \text{ m}\\[2ex] 1 \text{ cm} &: 400000 \text{ cm} \end{align*}}

The scale of the map is $\small{\displaystyle{\frac{1}{400000}}}$.

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