Similar Figures

1. Introduction to Similar Figures

A similar figure is two figures that have identical shapes. Congruent objects are objects that are precisely the same shape and size. In real life, for instance, both front wheels of a car, both hands of an individual, and so on are instances of congruent figures or objects. However, similar shape objects have the same shape but various sizes. The symbol ∼ is used to represent similarity.

2. Determining Scale Factors of Similar Figures

When two figures have the same shape but different sizes, they are referred to as similar figures in mathematics. For example, different sized photographs of a person, such as a stamp size, passport size, and so on, depict similar but incongruent objects. In geometry, two comparable shapes are those whose dimensions are equal or have a common ratio, but the size or length of their sides differ. Examples include similar triangles, similar rectangles, and similar squares. The scale factor refers to the common ratio. In addition, the corresponding angles are equal in size. You can also take Maths Assignment Help from experts.

A scale factor is the ratio of the corresponding sides of two similar objects. In order to find a scale factor between two similar figures, find two corresponding sides and write the ratio of the two sides. If someone starts with a smaller figure, their scale factor will be less than one. If someone starts with the larger figure, their scale factor will be greater than one. Two geometric figures are comparable if their corresponding angles and sides seem to be equal and proportional. A ratio is a fraction that compares two numbers. The scale factor is the ratio of the corresponding sides of two comparable geometric figures. Discover two corresponding sides, one on each figure, to determine the scale factor and to find the scale factor from one figure to the other, write the ratio of one length to the other. You can also simplify difficult fractions with the help of the Fraction Calculator tool.

3. Similar Figures: How is it useful?

When something is said to be similar to something else, it means that the two things share common characteristics. For instance, suppose two individuals brainstorm ways to solve a specific problem, and one tells the other the approach he would take. One person then tells the other that his approach is similar to the strategy he was considering. It means that their approaches to solving the problem are similar, but minor differences may be. In mathematics, the term "similar" refers to the fact that two figures have a similar shape. They can be of varying sizes, but they must be of the same shape. Take a look at this image of butterflies:

The butterflies are all the same shape, but they are all different sizes. As a result, the butterflies are comparable. Similar figures are very helpful in determining the sizes of items that are difficult to measure by hand in an indirect manner. Building heights, tree heights, and tower heights are common examples. Similar Triangles can also be used to determine the width of a river or lake.

4. Similar Figures Area and Volume

Similar figures are similar in shape but not always in size. In order to calculate a missing length, area, or volume on a reduction/enlargement figure, first determine the scale factor.

Similar areas:

We already know that if two shapes are similar, their corresponding sides and angles have the same ratio and are equal. The Area Scale Factor must be calculated when calculating a missing area.

Area Scale Factor (ASF) = (Linear Scale Factor)²

Example: The figures below are comparable. Determine the area that is missing.

LSF_enlargement = Big/Small = 16/8 = 2/1 = 2

ASF = 2² = 4

Area = 4 * 22 = 88cm²

Similar volumes:

The Volume Scale Factor must be calculated when calculating a missing volume.

Volume Scale Factor (VSF) = (Linear Scale Factor)3

Example: The vases below are comparable. Determine the volume that is missing.

LSF_reduction = Small/Bif = 6/8 = ¾

VSF = (3/4)³ = 27/64

Volume = 27/64 * 640 = 270ml (to nearest 1ml)

5. Properties of Similarity

Similar figures have a similar appearance but may not be the same size. Two objects are similar if they have the same shape but differ in size. When similar shapes are magnified or demagnified, they superimpose on each other. This property of similar shapes is known as "Similarity." Two triangles are similar if their angles are the same (corresponding angles) and their sides have the same ratio or proportion (corresponding sides). Similar figures may have various individual side lengths, but their angles must be equivalent, and their corresponding side length ratio must be the same. If two triangles are similar, it means that:

Triangle corresponding angle pairs are all equal; and
Triangles' corresponding sides are all proportional.

People use the "∼" symbol to represent the similarity. So, if two triangles are similar, we show it as △QPR ∼ △XYZ.

Similar triangles are triangles with equal corresponding angle pairs. As a result, equiangular triangles are identical. As a result, all equilateral triangles are comparable triangles. The triangles in the following image are similar, but their sizes are distinctive.

The following are the properties of similar figures:

Similar figures are similar in shape but differ in size;
Corresponding angles in similar figures are equivalent;
The ratio of corresponding sides of the similar figure is the same; and
The area ratio of a similar figure is the same as the square ratio of any pair of corresponding sides.

6. Most Common Similar Figures Examples

If two figures have the same shape, they are said to be similar. In more mathematical terms, two figures are comparable if their corresponding angles are congruent and their corresponding side length ratios are equal. The scale factor is the name given to this common ratio. The symbol "∼" is used to represent similarity.

Example 1: In the figure below, pentagon ABCDE ∼ pentagon VWXYZ.