CBSE Class 10 Mathematics Triangles Notes Set 11

Similar Triangles: Key Ideas

Two triangles are similar when their corresponding angles are equal, and then the corresponding sides are in the same ratio. Criteria: AA (two equal angles), SSS (three proportional sides) and SAS (one equal angle between proportional sides).
Basic proportionality theorem (Thales): a line parallel to one side of a triangle and meeting the other two sides divides them in the same ratio. Converse: if a line divides two sides in the same ratio, it is parallel to the third side.
Angle bisector theorem: the bisector of an angle of a triangle divides the opposite side in the ratio of the other two sides, BD/DC = AB/AC.

Areas and Special Results

The ratio of the areas of two similar triangles equals the square of the ratio of any pair of corresponding sides, altitudes or medians.
Triangles on the same base and between the same parallels have equal area; triangles of equal height have areas in the ratio of their bases.
In a right triangle the altitude to the hypotenuse makes two triangles similar to the whole triangle and to each other. This gives AD² = BD × DC, AB² = BD × BC and AC² = DC × BC.
Pythagoras: in a right triangle the square on the hypotenuse equals the sum of the squares on the other two sides. Its converse: if a² + b² = c², the angle opposite c is a right angle.
In a triangle with sides a, b, c, the angle opposite c is acute, right or obtuse according as c² is less than, equal to or greater than a² + b².

Useful Facts for Problems

The centroid divides each median in the ratio 2 : 1 measured from the vertex.
For a right triangle with legs a and b and altitude p on the hypotenuse c, pc = ab and 1/p² = 1/a² + 1/b².
In an equilateral triangle of side s the altitude is (√3/2)s and the circumradius is s/√3.
If an altitude AD of triangle ABC with AB = AC is drawn from A, then AB² − AD² = BD × CD for every point D on BC.
The medians AQ and BP of a right triangle ABC (right angle at C) satisfy 4(AQ² + BP²) = 5AB².