CBSE Class 10 Mathematics Triangles Notes Set B

Chapter : TRIANGLES

Key contents

Two figures are called similar if they have same shape, irrespective of the size.

1. Two triangles are similar if their corresponding angles are equal and corresponding sides are proportional.

2. (AAA similarity) If two triangles are equiangular, then the triangles are similar Cor: (AA similarity): If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar

3. (SSS similarity) If the corresponding sides of two triangles are proportional then they are similar

4. (SAS similarity) If in two triangles, one pair of corresponding sides are proportional and the included angles are equal, then the triangles are similar.

SOME IMPORTANT RESULTS AND THEOREMS

* Theorem no. 1: 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. (BASIC PROPORTIONALITY

THEOREM or BPT or THALES THEOREM).
* proof may be asked.

* Theorem no 2: If a line is drawn intersecting the two sides of a triangle such that it divides the two sides in the same ratio, then the line is parallel to the third side. (converse of BPT)

* proof may be asked

Theorem no 3: The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle.(angle bisector theorem)

Theorem no 4: In a triangle ABC, if D is the point on BC such that BD/DC = AB/AC, prove that AD is the bisector of angle A(converse)

Theorem no 5: The external bisector of an angle of a triangle divides the opposite side externally in the ratio of the sides containing the angle.

Theorem no 6: The line drawn from the mid point of one side of a triangle parallel to the other side bisects the third side.

Theorem no 7:The line joining the mid points of two sides of a triangle is parallel to the third side.

Theorem no 8:Prove that the diagonals of a trapezium divide each other proportionally.

Theorem no 9:If the diagonals of a quadrilateral divide each other proportionally , then it is a trapezium.

Theorem no 10:Any line parallel to the parallel sides of a trapezium divides the non parallel sides proportionally.

Theorem no 11:If two triangles are equiangular, prove that:

i) Ratio of the corresponding sides is the same as the ratio of the corresponding medians.

ii) Ratio of the corresponding sides is the same as the ratio of the corresponding angle bisector segments.

iii) Ratio of the corresponding sides is the same as the ratio of the corresponding altitudes.

Theorem no 12:: If one angle of a triangle is equal to one angle of another triangle and the bisectors of these equal angles divide the opposite side in the same ratio, prove that the triangles are similar

Theorem no 13: If two sides and the median bisecting one of these sides of a triangle are respectively proportional to the two sides and the corresponding median of another triangle, then the triangles are similar.

Theorem no 14:If two sides and a median bisecting the third side of a triangle are respectively proportional to the corresponding sides and the median of another triangle, then the two triangles are similar.

* Theorem no 15:The ratio of the areas of two similar triangles is equal to the ratio of the squares of the corresponding sides.
* proof may be asked

Theorem no 16: The areas of two similar triangles are in the ratio of

I) Squares of the corresponding altitudes

II) Squares of the corresponding medians

III) Squares of the corresponding angle bisectors.

Theorem no 17:if the areas of two similar triangles are equal then the triangles are congruent

5. * pythagoras theorem: in a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

* proof may be asked

* Converse of pythagoras theorem: In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite to the first side is a right angle.

* proof may be asked.

6. Some important results on pythagoras theorem:

1) Δ ABC is an obtuse angled triangle, obtuse angled at B. If ad is perpendicular to CB, prove that, AC2 = AB2 + BC2 +2 BC. BD

2) Δ ABC is an acute angled triangle, acute angled at B. If ad is perpendicular to CB, prove that, AC2 = AB2 + BC2 - 2 BC.

3) Prove that in any triangle, the sum of the squares of any two sides is equal to twice the square of half the third side together with twice the square of the median which bisects the third side. (APPOLONIUS THEOREM)

4) Prove that three times the sum of the squares of the sides of the triangle is equal to four times the sum of the squares of the medians of the triangle.