Area of Similar Triangles - Formula, Theorem, Proof, Examples

Similar triangles are defined as that type of triangles that have the same shape, but their sizes may differ. In equilateral triangles, the squares of any side length are examples of similar objects or triangles. In other words, if any two triangles are similar, then their corresponding angles are also congruent and corresponding sides are also in equal proportion. The similarity of triangles here given by the ‘~’ symbol.

Introduction

If any two triangles are similar, they always have the same ratio of corresponding sides and also equal pair of corresponding angles.

If any two or more figures have the same shape, but their sizes may differ, then these objects are known as similar figures.

Triangle is also known as a three-sided polygon. The condition is for only the similarity of triangles is;

• In a similar triangle the corresponding angles of both triangles are equal, and

• In a similar triangle corresponding sides of both triangles are always proportional to each other.
  
  

Properties Of Similar Triangles

• Reflexivity: It is defined when A triangle (△) is similar to itself.

• Symmetry: Symmetry is defined when △ ABC ∼ △ DEF, Then △ DEF ∼ △ ABC

• Transitivity: Transitivity is defined when △ ABC ∼△ DEF and△ DEF ∼△ XYZ, then △ ABC ∼△ XYZ

• Both triangles have the same shape but sizes may differ.

• Each pair of corresponding angles of a triangle are always equal

• The ratio of corresponding sides of a triangle is the same

Formulas

According to the definition, when two triangles are similar and their corresponding angles are congruent and their corresponding sides are always proportional. We can also find the dimensions of any one triangle with the help of another triangle. Now, let us consider ABC and XYZ are two similar triangles, then with the help of the given formulas, relevant angles and side lengths can be found out.

• \angle\:A=\angle\:B,\angle\:B=\angle\:C,\angle\:C=\angle\:Z

• AB\div\:XY=BC\div\:YZ=AC\div\:XZ

Similar Triangles Vs Congruent Triangles

The comparison of similar and congruent triangles is given below in the table.

Similar Triangles Theorems With Proofs

1. AA (or AAA) or Angle-Angle Similarity

2. SAS or Side-Angle-Side Similarity

3. SSS or Side-Side-Side Similarity

1)AA (or AAA) or Angle-Angle Similarity

If two angles of a triangle are equal to any two angles of another triangle, then the two triangles are similar to each other; this means the triangle has Angle-Angle Similarity.

2) SAS or Side-Angle-Side Similarity

If any two sides of a triangle are in the same proportion as of the two sides of another triangle, and that angle is inscribed by the two sides in both triangles are equal, then two triangles are said to be Side-Angle-Side similarity.

3) SSS or Side-Side-Side Similarity

If all three sides of a triangle are in proportion to the three sides of another triangle, then these two triangles are called to be Side-Side-Side similar.

Area Of Similar Triangles Theorem

According to the similar triangle, the theorem states that If any two triangles are similar, then the ratio of the area of both triangles will be proportional to the square of the ratio of corresponding sides.

To prove this theorem, Let us consider two similar triangles \triangle\:ABC and \triangle\:PQR

According to this theorem,

\frac{area\:of\triangle\:ABC}{area\:of\triangle\:PQR}

=\frac{\lgroup\:AB\rgroup^{2}}{\lgroup\:PQ\rgroup^{2}}

=\frac{\lgroup\:BC\rgroup^{2}}{\lgroup\:QR\rgroup^{2}}

=\frac{\lgroup\:CA\rgroup^{2}}{\lgroup\:RP\rgroup^{2}}

We know that,

Area of triangle = \frac{1}{2}\times\:base\times\:height

To find the area of \triangle\:ABC and \triangle\:PQR

, first draw the altitudes AD and PE from the vertex A and P of \triangle\:ABC and \triangle\:PQR respectively, as shown in the figure -

Now, area of ΔABC =

\frac{1}{2}\times\:BC\times\:AD

area of \triangle\:PQR

=\frac{1}{2}\times\:QR\times\:PE

The ratio of the areas of both triangles is given by:

\frac{area\:of\triangle\:ABC}{area\:of\triangle\:PQR}

=\frac{\frac{1}{2}\times\:BC\times\:AD}{\frac{1}{2}\times\:QR\times\:PE}

\frac{area\:of\triangle\:ABC}{area\:of\triangle\:PQR} =

=\frac{BC\times\:AD}{QR\times\:PE}

……………. (1)

Now in \triangle\:ABD and \triangle\:PQE

it can be seen that:

\angle\:ABC = \angle\:PQR (Since \triangle\:ABC \sim\triangle\:PQR )

\angle\:ABD = \angle\:PEQ (Since both the angles are 90°)

From AA criterion of similarity \triangle\:ADB \sim\triangle\:PEQ

\Rrightarrow\frac{AD}{PE}=\frac{AB}{PQ} …………….(2)

Since it is known that \triangle\:ABC \sim\triangle\:PQR

=\frac{AB}{PQ}=\frac{BC}{QR}=\frac{AC}{PR} …………….(3)

Now, substitute the value in equation (1), we get

\frac{area\:of\triangle\:ABC}{area\:of\triangle\:PQR} =

=\frac{AB}{PQ}\times\frac{AD}{PE}

Using equation (2), we have-

\frac{area\:of\triangle\:ABC}{area\:of\triangle\:PQR} =

=\frac{AB}{PQ}\times\frac{AB}{PQ} \frac{area\:of\triangle\:ABC}{area\:of\triangle\:PQR} =

=\lgroup\:\frac{AB}{PQ}\rgroup^{2}

Also from equation (3) we have-

\frac{area\:of\triangle\:ABC}{area\:of\triangle\:PQR} =

=\lgroup\:\frac{BC}{QR}\rgroup^{2}

=\lgroup\:\frac{CA}{RP}\rgroup^{2}

This proves that the ratio of the area of two similar triangles is always proportional to the squares of the corresponding sides of both triangles.