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

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.

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 \(\small{27.2 \text{ km}}\).

2. 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 \(\small{3.95 \text{ cm}}\).

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

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