SIMILARITY OF TRIANGLES

SIMILARITY OF TRIANGLES  CLASS 10 CHAPTER 6

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 similar.
• 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.

For Explanation of this theorem  Click Here

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

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 \[\frac{AB}{AC}=\frac{BD}{DC}\]

SIMILARITY OF TRIANGLES

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