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Class 10 Maths notes for similar triangles



Similar Figures


Two figures having the same shape but not necessarily the same size are called similar figures.
Similar Triangles
Examples:
All circles are similar
All square are similar
All equilateral triangles are similar
All the congruent figures are similar but the converse is not true

Similar Polygons


Two polygons with same number of sides are said to be similar, if
(i) their corresponding angles are equal and
(ii) their corresponding sides are in proportion (or are in the same ratio).

Basic Proportionally Theorem (or Thales Theorem)


If a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio.
Similar Triangles
In the ΔABC , if DE || BC,
Similar Triangles
Proof:
Construction: Join BE and CD. Draw DM ⊥ AC and EN ⊥ AB

Consider ΔADE and ΔBDE.

ar(ΔADE) = ½ x AD x EN

ar(ΔBDE) = ½ x DB x EN

ar( ? ADE) /ar( ? BDE)    = ( 1/ 2 x AD x DM )/(1/ 2 x DB x DM)         = AD/ DB  ...........(1)

Consider ΔADE and ΔDEC

ar(ΔADE) = ½ x AE x DM

ar(ΔDEC) = ½ x EC x DM

ar( ? ADE) /ar( ? DEC)    =   (1/ 2 x AE x DM)/ (1/ 2 x EC x DM)     = AE/ EC  ..........(2)

Consider ΔBDE and ΔCED

ar(ΔBDE) = ar(ΔCED) .........(3)

[Since ΔBDE and ΔCED are on the same base, DE, and between the same parallels, BC and DE.]

Therefore
AD/DB  AE/EC   [From (1), (2) and (3)]
Converse of Basic Proportionality Theorem:
If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

Criteria for Similarity of Triangles


Now triangle is also type of polygon and we already know the similarity criteria for that. So Two triangles are said to be similar, if
(i) their corresponding angles are equal and
(ii) their corresponding sides are in proportyion (or are in the same ratio). Similar Triangles
Corresponding angles are equal
∠A=∠D,∠B=∠E,∠C=∠F
Corresponding sides are in proportyion (or are in the same ratio).
AB/DE = BC/EF = AC/DF
Symbolically, we write the similarity of these two triangles as
Δ ABC ~ Δ DEF
The symbol ‘~’ stands for ‘is similar to’. Recall that you have used the symbol ‘≅’ for ‘is congruent to’ in Class IX
We should keep the letters in correct order on both sides

Different Criterion for similarity of the triangles

(i) AA or AAA Similarty Criterion:
Similar Triangles
(ii) SSS Similar Criterion:

We may recall that the below two conditions
(i) corresponding angles are equal and
(ii) corresponding sides are in the same ratio
are required  for two polygons to be similar
However, on the basis of last two SSS and AAA criterio you can now say that in case of similarity of the two triangles, it is not necessary to check both
the conditions as one condition implies the other.
(iii) SAS Similar Criterion:
Similar Triangles
Examples
1) if PQ || RS, prove that Δ POQ ~ Δ SOR
Similar Triangles
Solution : Given PQ || RS
So, ∠ P = ∠ S (As per Alternate angles)
and ∠ Q = ∠ R
Also, ∠ POQ = ∠ SOR (As they are Vertically opposite angles)
Therefore, Δ POQ ~ Δ SOR (AAA similarity criterion)
2) The side lengths of  ΔPQR are 16, 8, and 18, and the side lengths of ΔXYZ are 9, 8, and 4. Find out if the triangle is similar
Solution
For these questions , it is recommended to compare the ratio of largest side and shortest side and then remaining side
So 18/9=2 ( Longest side)
8/4=2 ( Shortest side)
16/8=2 ( Remaining side)
Since all ratio's are equal , triangles  are similar
3) In ΔABC, ∠A = 22° and ∠B = 68°. In ΔDEF, ∠D = 22° and ∠F = 90°.
Solution:
For ΔABC,three angles are
∠A = 22° and ∠B = 68°
∠C= 180 - ( ∠A+ ∠B) = 90
For ΔDEF,three angles are
∠D = 220 and ∠F = 900
∠E= 180 - ( ∠D+ ∠F) = 68
So from AAA similarity criterion, triangles are similar
ΔABC ~ ΔDEF

Areas of Similar Triangles


The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
.Similar Triangles
Given Δ ABC ~ Δ DEF ar (ABC)/ar (DEF) =(AB/DE)2 = (BC/EF)2 = ( AC/DF)2
Also
The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding altitudes.
The ratio of the areas of two similar triangles is equal to the sum of the squares of their corresponding angle bisectors.
Example
1) Let Δ ABC ~ Δ DEF and their areas be, respectively, 64 cm2 and 121 cm2. If EF = 15.4 cm, find BC.
Solution:
We know that
ar( ABC)/ar( DEF)= (BC/EF)2
64/121=(BC/15.4)2
BC= 11.2  cm

Pythagoras Theorem:


In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Similar Triangles
AC2 = AB2 +BC2

Converse of Pythagoras Theorem:


In a triangle, if square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.
If a perpendicular is drawn from the vertex of the right angle of a right trianglr to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle and similar to each other.
i. e., if in triangle ABC,
<B = 90 and BD - AC, then
Similar Triangles
(i)  ADB ~  ABC
(ii)  BDC  ~  ABC
(iii) ADB ~  BDC  RHS Similarity Criterion:
If in two right triangles, hypotenuse and one side of one triangle are proportional to the hypotenuse and one side of the other triangle, then the two triangles are similar by RHS similarity criterion.


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