Similar Triangles: Definition, Example and Formula, Properties

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Similar triangles are the triangles which have the same shape, but their sizes may vary. All equilateral triangles, squares of any side length are examples of similar objects. In other words, if two triangles are similar, then their corresponding angles are congruent and corresponding sides are in equal proportion. We denote the similarity of triangles here by symbol.

Similar Triangles Definition

If two or more figures have the same shape but their sizes are different then such objects are called Similar figures. Consider a hula hoop and wheel of a cycle, the shapes of both these objects are similar to each other as their shapes are the same.

Similar Triangles Definition

Similar Triangles Definition

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In the figure given above, two circles with radius R and r respectively are similar as they have the same shape, but necessarily not the same size. Thus, we can say that

It is to be noted that, two circles always have the same shape, irrespective of their diameter. Thus, two circles are always Similar.

Triangle is the smallest three-sided polygon. The condition for the similarity of triangles is;

i) Corresponding angles of both the triangles are equal, andii) Corresponding sides of both the triangles are in proportion to each other.

Similar Triangle Example

Similar Triangles Example

Similar Triangles Example

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In the given figure, two triangles and are similar only if,

i) ii)

Hence if the above-mentioned conditions are satisfied, we can say that

It is interesting to know that if the corresponding angles of two triangles are equal then such triangles are known as equiangular triangles. For two equiangular triangles we state the Basic Proportionality Theorem (better known as Thales Theorem) as follows:

For two equiangular triangles, the ratio of any two corresponding sides is always the same.

Properties

  • Both have the same shape, but sizes are different

  • Each pair of corresponding angles are equal

  • The ratio of corresponding sides is the same

Similar Triangle Formula

According to the definition, two triangles are similar if their corresponding angles are congruent and corresponding sides are proportional. Hence, we can find the dimensions of one triangle with the help of another triangle. If ABC and XYZ are two similar triangles then by the help of below-given formulas or expression we can find the relevant angles and side length.

Once we have known all the dimensions and angles of triangles, then find the area of similar triangles.

Similar Triangles Theorems with Proofs

Let’s solve the problems based on similar triangles along with the proofs for each.

AA (Or AAA) or Angle-Angle Similarity

If any two angles of a triangle are equal to any two angles of another triangle then the two triangles are similar to each other.

From the figure given above, if and then

From the result obtained, we can easily say that,

and

SAS or Side-Angle-Side Similarity

If the two sides of a triangle are in the same proportion of the two angles of another triangle, and the angle inscribed by the two sides in both the triangle are equal, then two triangles are said to be similar.

Thus, if and then

From the congruency, we can state that,

and

SSS or Side-Side-Side Similarity

If all the three sides of a triangle are in proportion to the three sides of another triangle then the two triangles are similar.

Thus, if

From this result, we can infer that-

Similar Triangles Problem

Let us go through an example to understand it better.Q.1: In the length of the sides are given as , and Also . Find

Similar Triangles Problem

Similar Triangles Problem

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Solution:

In and is common and (corresponding angles)

β‡’ (AA criterion for similar triangles)

Here given,

β‡’

Put the values,

β‡’

Cross Multiplication,

Example 2: Show that the two triangles given beside are similar and calculate the lengths of sides and .

Similar Triangles Problem

Similar Triangles Problem

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Solution:

and , (because )

Therefore, the two triangles and are similar. Consequently: