Similarity of Triangles - GMS - Learning Simply
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Similarity of Triangles

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Similarity of Triangles

Triangle is a polygon which has three sides and three vertices. Triangles having same shape and size are said to be congruent. The similarity of triangles uses the concept of similar shape and finds great applications. Triangles are said to be similar if:

a. Their corresponding angles are equal.

b. Their corresponding sides are in the same ratio.

Theorems and Proofs

Theorem: If a line divides any two sides of a triangle in the same ratio, then the line is said to be parallel to the third side.

Similarity Of Triangles –Theorems

Let ABC be a triangle with sides AB, AC and BC. DE divides any two sides of the triangle in the same ratio.

Given: Line divides a triangle in the same ratio. Thus,

ADDB = AEEC

Proof:

ADDB = AEEC (Given)—-(1)

Let us assume that DE is not parallel to BC. Now we draw DE’ which is assumed to be parallel to BC. So,

ADDB = AEEC (Property of similar triangles)—-(2)

Therefore, from (1) and (2)

AEEC = AEEC




Now we add 1 to both sides,

AEEC + 1 = AEEC + 1

 AEEC + ECEC = AEEC + ECEC

 AE + ECEC = AE+ECEC

According to the figure AE+EC = AC and AE+EC = AC, substituting these values in the equation above:

ACEC = ACEC

This directly implies that EC = EC and E = E meaning that they are the same point.

Hence DE is parallel to BC. This proves the similarity of triangles.

Examples

Let us take an example to observe the property of similarity of triangles:

Illustration 1:PQRS is a trapezium with PQ parallel to RS. The point X and Y are on the non-parallel sides PS and QR respectively such that XY is parallel to PQ. Show that PXXS = QYYR.

Similarity Of Triangles –Theorems

Solution: Let us first join PR in order to intersect XY at Z.

Similarity Of Triangles –Theorems

PQ || RS and XY || PQ (given)

So, XY || RS (lines parallel to same line are parallel to each other)

In  PSR,

XZ || SR (as XY || SR)

So, PXXS = PZZR …….. (3)

Similarly, from  PRQ

RZZP = RYYQ

PZZR = QYYR …………. (4)

From equation (3) and (4),

PXXS
 = QYYR<





About the Author

At the helm of GMS Learning is Principal Balkishan Agrawal, a dedicated and experienced educationist. Under his able guidance, our school has flourished academically and has achieved remarkable milestones in various fields. Principal Agrawal’s visio…

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