## What are similar figures

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Similar figures are 2 figures with the exact shape. Objects having the exact same size and shape are called **congruent objects**. For instance, a person’s two hands or the 2 front wheels on a vehicle are a couple of examples of congruent objects. However, objects can be of a similar shape yet have **varied sizes**. Use the \(∼\) **symbol **to represent similarity.

Some types of figures or geometrical shapes are similar all the time. Think about **circles**. Even if the circle’s radius changes **all the time**, the shape still remains the same. So, you can state that **all circles **having various radii will be **similar **to one another. The figure shown below signifies concentric circles whose radii are not the same, however, they are all still **similar**. Though the circles are shaped the same, the circles’ **sizes **aren’t the same. So, these circles **aren’t** congruent.

### Similar Figures Defined

In Mathematics, whenever 2 figures are shaped the **same **yet have different **sizes**, they are known as similar figures. For instance, people’s photos having a different size (such as passport, stamp, etc.) are similar objects yet they aren’t congruent. With geometry, 2 similar shapes like similar squares, rectangles, and triangles, are shapes that have dimensions with an equal or a common ratio yet their length or size aren’t the same. A common ratio is known as a **scale factor**. Additionally, the parallel angles are exactly the same size.

If 2 figures are said to be similar they get signified by using the ‘\(∼\)’ **symbol**. So, if there are 2 similar triangles, we’ll call them ABC and PQR, then they‘re signified via: \(∆ABC\) \(∼\) \(∆PQR\).

But if 2 **triangles** are similar to one another, their parallel sides must be in proportion. So,

- \(\frac{AB}{PQ} \ = \ \frac{BC}{QR} \ = \ \frac{AC}{PR}\)
- \(∠A \ = \ ∠P, \ ∠B \ = \ ∠Q, \ ∠C \ = \ ∠R\)

### Area and Volume of Similar Figures

If 2 figures are similar, their **parallel **sides are proportional. Plus, when the ratio of the triangles’ sides is the same.

So, if you use the ratio of the triangles’ **surface **areas, this is going to be the same as the **square **of the ratio of the side. And the ratio of the **volume **of 2 similar figures is going to be the same as the **cube **of the ratio of the sides’ length.

**Note**: It is referring to the ratios, not the figures’ surface area and volume.

So, dependent on the above statements, the **scale factors** of the area and volumes’ scale factors can be shown like this:

\(SF_{A} \ = \ SF^2\)

\(SF_{V} \ = \ SF^3\)

in which \(SF_{A}\) is the **surface **area’s scale factor and \(SF_{V}\) is the **volume**’s scale factor.

### Examples of Similar Figures

Look at the below figures. They all are similar to one another, since their shapes are the **same**, yet they’re **not **considered to be congruent. You have to remember that when someone says there are similar figures, it is based on **shape **alone, **not **counting the sizes of the objects. So, it can be concluded that while all the congruent figures will be similar, all the similar figures **aren’t** going to be congruent.

If you take all the similar figures, you are able to state that for **any **n-sided polygon, its **inclination **angles of the line segments are the same all the time, no matter what size the figure is. So, for any 2 n-sided polygons with the same number of sides, one can state they’re similar when:

- Parallel
**angles**of both polygons are**equal**, as well as - Parallel
**sides**of both polygons will be in the exact**same ratio**.

So, **congruence **is a distinct example of similarity whenever the ratio of the sides of a figure is \(1\). Which means, whenever 2 similar figures are congruent, the **corresponding **length of their sides is equal since the ratio of the parallel sides is \(1\).

### Exercises for Similar Figures

**1)** \(33 : 36 = 11 : x \)

**2) ** \(63 : 84 = 3 : x \)

**3) **\(69 : 87 = 23 : x \)

**4) **\(30 : 60 = 1 : x \)

**5) **\(27 : 30 = 9 : x \)

**6) ** \(98 : 105 = 14 : x \)

**7) **\(52 : 108 = 13 : x \)

**8) **\(20 : 40 = 1 : x \)

**9) **\(16 : 64 = 1 : x \)

**10) **\(32 : 40 = 4 : x \)