CBSE Class 10 Mathematics Triangles Notes Set 12

Similar Triangles and Pythagoras Theorem

Basic proportionality (Thales) theorem: a line drawn parallel to one side of a triangle to meet the other two sides divides them in the same ratio, so \( \frac{AD}{DB}=\frac{AE}{EC} \). The converse also holds: if a line divides two sides in the same ratio it is parallel to the third side. Consequences: a line through the mid-point of one side parallel to another bisects the third side; the line joining mid-points of two sides is parallel to the third side and half as long; the diagonals of a trapezium divide each other in the same ratio.
The internal bisector of an angle of a triangle divides the opposite side in the ratio of the other two sides; the external bisector does so externally.

Similarity and Areas

Two triangles are similar when their angles are equal (AA test) or their sides are proportional. The ratio of areas of similar triangles equals the square of the ratio of corresponding sides, and also the square of the ratio of corresponding altitudes, medians or angle bisector segments. Example: sides in ratio 3 : 5 give areas in ratio 9 : 25; midpoint triangle DEF has one quarter of the area of ABC.

Pythagoras Theorem and Applications

In a right triangle \( AC^2=AB^2+BC^2 \); the converse says a triangle whose sides satisfy this has a right angle opposite the longest side (7, 24, 25 is one example). The altitude to the hypotenuse creates two triangles similar to the original, giving \( CD^2=AD\cdot BD \), and the product of the legs equals hypotenuse times that altitude.
Results for other triangles: for an obtuse angle at B, \( AC^2=AB^2+BC^2+2BC\cdot BD \); for an acute angle, \( AC^2=AB^2+BC^2-2BC\cdot BD \) (D the foot of the perpendicular from A). For a median m to side a, \( b^2+c^2=2m^2+\frac{a^2}{2} \), and three times the sum of the squares of the sides equals four times the sum of the squares of the medians.