Similarity of Triangles, SSS Similarity Rule, SAS Similarity Rule, AAA Similarity Rule, AA Similarity Rule, B.P.T. Pythagoras Theorem
Similar figures:-
Two figures which looks same but not necessarily having the same size are called similar figures.
Similarity of Triangles
Two triangles are similar if their corresponding sides are
proportional and angles are equal.
For example :- Two same photos of different size are similar to each other.
- All equilateral triangles are similar.
- All squares are similar to each other.
- All congruent figures are similar to each other, but all
similar figures may or may not be congruant.
- Similarity of two figures are shown by using the symbol "〜"
- A polygon is said to be similar if its corresponding angles are equal and sides are proportional.
Similarity conditions of triangles :-
Triangles can be made similar with the help of four
conditions, SSS, SAS, AAA, AA conditions
Explanation of all conditions
Side – Side
- Side (SSS) Similarity Rule :-
Two triangles are said to be similar by SSS similarity rule if their corresponding sides are proportional.
Side –
Angle - Side (SAS) Similarity Rule :-
Two triangles are said to be similar by SAS similarity rule
if their two corresponding sides are proportional to each other and
corresponding angles included between the proportional sides are equal.
Angle –
Angle - Angle (AAA) Similarity Rule :-
Two triangles are said to be similar by AAA similarity rule If
three angles of one triangle are equal to the three angles of other triangle.
Angle –
Angle Similarity Rule :-
Two triangles are said to be similar by AA similarity
rule If two angles of one triangle are
equal to the two angles of other triangle.
Basic
Proportionality Theorem (B.P.T.) or Thales Theorem :-
If a line is drawn parallel to one side of triangle to
intersect the other two sides at two distinct points, then other two sides are
divided into same ratio.
Important
Notes:-
- If two triangles are similar then their corresponding sides,
medians, altitudes and perimeters are proportional to each other.
- If two triangles are similar then ratio
of areas of two similar triangle is equal to the ratio of square of their
sides, their medians, their altitudes.
- In a right angled triangle if a perpendicular is drawn from
the right angle to the hypotenuse then triangles on either side of the
perpendicular is similar to each other and to the whole triangle.
Important Result :
Internal bisector of angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle. In Triangle ABC, if AD is the bisector of ∠A then \[\frac{AB}{AC}=\frac{BD}{DC}\]
Topics Deleted from CBSE syllabus Theorem on Area of two similar triangles:-
Ratio of areas of two similar triangles is equal to the ratio of square of their corresponding sides.
For Explanation of this theorem Click Here
Pythagoras Theorem:-
In right angled triangle the square of the longest side is equal to the sum of square of the other two sides.
For Explanation of this theorem Click Here
Converse of Pythagoras theorem :-
If square of one side of a triangle is equal to the sum of square of the other two sides then angle opposite to the longest side is equal to the right angle.
For Explanation of this theorem Click Here
SIMILARITY OF TRIANGLES
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