Class 10 – Geometry

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Introduction: Observe the following figures

By observation

i), ii) are of the same shape but different in size iii), iv) are of the same shape and same size v), vi) are different in shape as well as size.

Similar figures: Two figures are said to be similar if they have the same shapes although they may differ in sizes.  The following pairs of figures are similar.

 

Similar triangles: If two triangles are similar then their

i)               Corresponding angles are equal, and

ii)              Corresponding sides are in proportion.

 

Here consider

So,

The symbol ‘~’ means “is similar to”.

 

Corresponding sides: Sides opposite to equal angles in similar triangles are known as corresponding sides and they are proportional.

.

Therefore, AB and DE are corresponding sides as they are opposite to and respectively.

Similarly, BC and EF are a pair of corresponding sides.  AC and DF are also a pair of corresponding sides.

Thus, , as corresponding sides of similar triangles are proportional.

Corresponding angles: Angles opposite to proportional sides in similar triangles are known as corresponding angles.

as they are opposite to corresponding sides BC and EF respectively.

Similarly, .

Congruency and similarity of triangles:Congruency is particular case of similarity.  In both the cases, three angles of one triangle are equal to the three corresponding angles of the other triangle.  But in similar triangles the corresponding sides are proportional, while in congruent triangles the corresponding sides are equal.

Where k is the constant of proportionality or the scale factor of size transformation.

Therefore, in congruent triangles the constant of proportionality between the corresponding sides is equal to one.  Thus, congruent triangles have the same shape and size while similar triangles have the shape but not necessarily the same size.

Congruent triangles are always similar, but similar triangles are not necessarily congruent.

Description: http://images.clipartpanda.com/bloc-clipart-note-md.pngTriangles that are similar to the same triangle are similar to each other.

Criteria of similarity between triangles:

1. SAS criterion of Similarity:If two triangles have their corresponding angles equal and the sides including between a pair of angles are proportional, then the triangles are similar.

                                   

2. AA or AAA Criterion of similarity: If two angles of one triangle are equal to two corresponding angles of the other, then the triangles are similar.

If in two triangles, two angles of one are equal to two angles of the other, then the third angle of the first triangle is also equal to the third angle of the other because the sum of the three angles in a triangle is 1800.

Thus, similar triangles are equiangular.

3. SSS criterion of similarity: If in two triangles, three sides of one are proportional to the three sides of the other, then the triangles are similar.

Theorem:In a right-angled triangle, if a perpendicular is drawn from the right angled vertex to the hypotenuse, the triangles on each side of it are similar to the whole triangle and to one another.

Given: Let ABC be a triangle in which angle

To prove:

Proof:


(Given)

(Common angle)

 Therefore ( By AA criterion of similarity) . . . . .(1)

(Given)

(Common angle)

Therefore ( By AA criterion of similarity) . .. .. .. (2)

 from 1, 2 .

Hence Proved.

Basic proportionality theorem:

Theorem:A line drawn parallel to one side of a triangle divides the other two sides in the same ratio (proportion).

Given:In ÄABC,D and E are points on AB and AC respectively, such that DE BC

To prove:

Proof:

 

( Common angle)

( Corresponding angles)

( AA criterion of similarity)

 (Corresponding sides of similar triangles are proportional)

( By subtracting 1 from both sides)

(by reciprocating term in above step).

Hence Proved.

Converse of Basic Proportionality Theorem:

Theorem:The line dividing two sides of a triangle proportionally is parallel to the third side.

Given:ΔABC, D and E are points on AB and AC respectively, such that

To Prove:

Proof:

(Given)

(Taking reciprocals of both sides)

(By adding 1 both sides)

In

(From statement 1)

(Common angle)

(By SAS criterion of similarity.

(Corresponding angles of similar triangles are equal)

(Corresponding angles are equal)

Hence Proved.

An important application of Basic Proportionality Theorem: The internal bisector of an angle of a triangle divides the opposite side in the ratio of the sides containing the angle. (Vertical angle bisector theorem).

Given: AD is the internal bisector of

To prove:

Construction: 

Proof:

(Given) integral bisector    . . . . (1)

()

()

(from 1)

 AE = EC (opposite sides of equal angles of ) . . . (2)

()

(By statement 2)

Hence Proved.

Description: http://images.clipartpanda.com/bloc-clipart-note-md.pngThe above proposition is true for external division also.

Converse of the above proposition is also true.  So, if D is a point on BC such that
BD : DC = AB : AC then AD bisects the angle BAC internally or externally.

Area of similar triangles: The ratio of the areas of two similar triangles has relation with the ratio of the corresponding sides. The ratio of the areas of two similar triangles is the square of the ratio of their corresponding sides.

Theorem:

Statement:The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.

Given:

So,

Also,

To prove:

Construction: Through A draw AP  BC and through D draw DQ  EF.

Proof:

Thus,

Putting the value of  from (iv) in (i), we get

Similarly, it can also be prove that

and

From (v), (vi) and (vii), we obtain

Example: Prove that the ratio of the areas of two similar triangles is equal to the ratio the squares of their corresponding: i) altitudes           ii) angle bisectors

Solution:

i) altitudes

Given:  

To prove:

Proof:

ii) Angle bisectors

Given:

AM is the bisector of and DN is the corresponding bisector of

To prove:

Proof:

Description: http://www.aids.harvard.edu/img/news/spotlight/newsletter_layout/logo.gif
Given below is the proof given in two ways. Let us check the relevancy of the proof for similarity.

Some students solve it as: The two triangles are similar as

Correct statements is: Here  but including angles  and  are not equal. So the given triangles are not similar.