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Triangles

Similarity of Triangles

To understand the meaning of similarity, imagine the Taj Mahal. Now imagine a mini version of it. Did you get it? The mini version is just a scaled-down version of the actual monument. The shape remains the same just the size changes. Same is the situation with the Triangles. Here is how to check the similarity of triangles.

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Similarity of Triangles

Two triangles are said to be similar when they have two corresponding angles congruent and the sides proportional.

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In the above diagram, we see that triangle EFG is an enlarged version of triangle ABC i.e., they have the same shape. The same shape of the triangle depends on the angle of the triangles.

\( \angle ABC = \angle EGF,  \angle BAC= \angle GEF, \angle EFG= \angle ACB \)

The area, altitude, and volume of Similar triangles are in the same ratio as the ratio of the length of their sides.

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Download Triangles Cheat Sheet PDF

Triangle Cheat sheet

Properties of Similar Triangles:

(A) Reflexivity: A triangle () is similar to itself

(B) Symmetry: If  ABC ∼  DEF, Then  DEF ∼  ABC

(C) Transitivity: If ABC ∼ DEF and DEF ∼ XYZ, then  ABC ∼ XYZ

Tests to prove that a triangle is similar

Angle-Angle Similarity(AA)

If two corresponding angles of the two triangles are congruent, the triangle must be similar.

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Side-Side-Side Similarity(SSS)

If the corresponding sides of the two triangles are proportional the triangles must be similar.

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Side Angle Side Similarity (SAS)

If two sides of two triangles are proportional and they have one corresponding angle congruent, the two triangles are said to be similar.

Theorem

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides

Given: A(ABC)~A(△PQR)

To Prove: A(ABC)/A(△PQR)=AB2/PQ2

Construction: Construct seg AM perpendicular side BC and seg PN perpendicular side QR

Proof:

A(ABC)/A(PQR)=(BC)(AM)/(QR)(PN)                        …(1)

A(ABC)~A(△PQR)

Therefore, \( \angle B = \angle Q \)

AB/PQ=BC/QR=AC/PR                                                           …(2)

In triangle ABM and triangle PQN,

\( \angle ABM=\angle PQN \)                                                …(from 1)

\( \angle AMB= \angle PNQ \)                                               …(each side is a right angle)

Therefore, △ ABM ~ PQN                                                   …(AA test of similarity)

AB/PQ=AM/PN       (c.s.s.t)                                                     …(3)

A(ABC)/A(△PQR)=[(BC)/(QR)][(AB)/(PQ)]                 …(from 1, 2 and 3)

A(ABC)/A(△PQR)=[(AB)/(PQ)][(AB)/(PQ)]                  …(from 3)

A(ABC)/A(△PQR)=AB²/PQ²

Similarly, we can show that,

A(ABC)/A(△PQR)=BC²/QR²=AC²/PR² 

Hence, we have proved that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

Solved Examples

Question 1: It’s given that  DEF ~ MNK. If DE= 5 and MN=6, find A(DEF)/A(MNK)

Answer : A(△DEF)/A(△MNK)=DE²/MN²  (areas of similar triangles)

=5²/6² =25/36

Question 2: How can one find if the triangles are similar?

Answer: The triangles are similar if:

  • All the angles of triangle are equal
  • The corresponding sides of triangle are in the same ratio

Question 3: Explain the SAS similarity theorem?

Answer: The SAS Similarity Theorem states that one triangle’s angle is congruent to another triangle’s corresponding angle such that the lengths of the sides, as well as these angles, are in proportion, then one can say that the triangles are similar.

Question 4: What is meant by SSS similarity?

Answer: According to SSS similarity, triangles are similar if one triangle’s three sides are in the same proportion to the other triangle’s corresponding sides. SSS is one of the three ways for testing the similarity of the triangles.

Question 5: What is meant by AA similarity theorem?

Answer: AA similarity postulate means that two triangles shall be similar if they have two corresponding angles such that they are equal or congruent in measure. Using this postulate, there will be no need to show that all three corresponding angles belonging to two triangles are equal for the purpose of proving that they are similar.

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