Area of Similar Triangles Explained with Formula and Proof

The area of two similar triangles suggests that if two triangles stand similar to each other, then the ratio of areas of similar triangles will be proportional to the square of the ratio of corresponding sides of similar triangles. This proves that the ratio of the area of both the similar triangles is proportional to the squares of the corresponding sides of the two similar triangles. The similarity of triangles is denoted by the symbol ‘~’.

Properties of Area of Similar Triangles

• The two similar triangles have the same shape but may differ in sizes.

• The ratio of corresponding sides of similar triangles is the same.

• Each pair of corresponding angles of similar triangles are equal.

Formulas of Area of Similar Triangles

As per the definition, two triangles are known to be similar if their corresponding sides are proportional and corresponding angles are congruent. Thus, we can determine the dimensions of one triangle using another triangle. If PQR and XYZ are two similar triangles, then using the below-given formulas, we can simply identify the relevant side lengths and angles.

 ∠P = ∠X, ∠Q = ∠Y and ∠R = ∠Z

PQ/XY = QR/YZ = PR/XZ

Once we get familiar with all the angles and dimensions of triangles, it is easy to identify the area of similar triangles.

Similar Triangles and Congruent Triangles

Below is the comparison of similar triangles and congruent triangles in the tabular form.

Similar Triangles Theorems with Proofs

There are various theorems that we will now learn. These are basically used to solve the problems surrounded on similar triangles along with the proofs for each.

1. Angle-Angle Similarity or (AAA)

By the principle of AAA, it suggests that if any two angles of a triangle are in equivalence to any two angles of another triangle, then the two triangles will be similar to each other.

From the figure given below, if ∠ A = ∠D and ∠C = ∠F then ΔABC ~ΔDEF.

From the outcome we attained, we can easily conclude that,

AB/DE = BC/EF = AC/DF

And ∠B = ∠Y

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2. Side-Side-Side Similarity (SSS)

By the postulation, if all the three sides of a triangle are given in a proportion to the three sides of another triangle, then the two triangles are said to be similar.

Thus, if AB/DE = BC/EF = AC/DF then ΔABC ~ΔDEF.

From the above outcome we obtained, we can come to the conclusion that-

∠A = ∠D, ∠B = ∠E and ∠C = ∠F

3. Side-Angle-Side Similarity or (SAS)

By the postulation, it is implied that if the two sides of a triangle or a similar object are in the same proportion of the two sides of the another triangle, and the angle carved out by the two sides in both the triangles are equivalent to one another, then two triangles are said to be similar.

Therefore, if ∠A = ∠D and AB/DE = AC/DF then ΔABC ~ΔDEF.

From the principle of congruence,

AB/DE = BC/EF = AC/DF

and ∠B = ∠E and ∠C = ∠F

Solved Examples

Let's consider an example to understand the similar triangles and congruence in a better way.

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Example:

In the ΔABC the length of the sides are given as AP = 10 cm , PB = 15 cm and BC = 30 cm. Also the side PQ||BC. Determine the PQ.

Solution:

In the given ΔABC and ΔAPQ,

∠PAQ is common and ∠APQ = ∠ABC (using the corresponding angles)

⇒ ΔABC ~ ΔAPQ (By the principle of AAA criterion for similar triangles)

⇒ AP/AB = PQ/BC

⇒ 10/20 = PQ/30

⇒ PQ = 20/2 cm

⇒ PQ = 10 cm

Example:

Check if the two triangles namely ΔABC and ΔDEF, are similar. The two angle given in triangle ABC are ∠A = 40° and ∠B = 70° while for triangle DEF are ∠D=60° and ∠F=80°.

Solution: 

In triangle ABC, using angle sum property;

∠A + ∠B + ∠C = 180°

40° + 70° + ∠C = 180°

110° + ∠C = 180°

Subtract both sides by 110°.

∠ C= 70°

Again in triangle DEF, ( by the criterion of angle sum property)

∠D + ∠E + ∠F = 180°

∠60° + ∠E + ∠80°= 180°

∠ 140° + ∠E = 180 °

Now, Subtracting both sides by 140°

we get

∠ E = 40°

Since,∠A = ∠ E = 40° and ∠C = ∠ F= 70°

Thus, by Angle-Angle (AA) rule, 

ΔABC~ΔDEF.