Similar Triangles, Congruent Figures and Basic Proportionality Theorem - Triangles, CBSE, Class 10, PDF Download

SIMILAR TRIANGLES

INTRODUCTION

In earlier classes, you have learnt about congruence of two geometric figures, and also some basic theorems and results on the congruence of triangle. Two geometric figures having same shape and size are congruent to each other but two geometric figures having same shape are called similar. Two congruent geometric figures are always similar but the converse may or may not be true.
All regular polygons of same number of sides such as equilateral triangle, squares, etc. are similar. All circles are similar.
In some cases, we can easily notice that two geometric figures are not similar. For example, a triangle and a rectangle can never be similar. In case, we are given two triangles, they may appear to be similar but actually they may not be similar. So, we need some criteria to determine the similarity of two geometric figures. In particular, we shall discuss similar triangles.

HISTORICAL FACTS

EUCLID was a very great Greek mathematician born about 2400 years ago. He i scalled the father of geometry because he was the first to establish a school of mathematics in Alexandria. He wrote a book on geometry called "The Elements" which has 13 volumes and has been used as a text book for over 2000 years. This book was further systematized by the great mathematician of Greece like Thales, Pythagoras, Pluto and Aristotle.
Abraham Lincoln, as a young lawyer was of the view that this greek book was a splendid sharpner of human mind and improves his power of logic and language.
A king once asked Euclid, "Isn't there an easier way to understand geometry"

Euclid replied : "There is no royal-road way to geometry. Every one has to think for himself when studying."

THALES (640-546 B.C.) a Greek mathematician was the first who initiated and formulated the theoretical study of geometry to make astronomy a more exact science. He is said to have introduced geometry in Greece. He is believed to have found the heights of the pyramids in Egypt, using shadows and the principle of similar triangles. The use of similar triangles has made possible the measurements of heights and distances. He proved the well-known and very useful theorem credited after his name : Thales Theorem.

CONGRUENT FIGURES
Two geometrical figures are said to be congruent, provided they must have same shape and same size.
Congruent figures are alike in every respect.
Ex. 1. Two squares of the same length.
2. Two circle of the same radii.
3. Two rectangles of the same dimensions.
4. Two wings of a fan.
5. Two equilateral triangles of same length.

SIMILAR FIGURES
Two figures are said to be similar, if they have the same shape. Similar figures may differ in size. Thus, two congruent figures are always similar, but two similar figures need not be congruent.
Euclid : Father of Geometry
(about 300 B.C. Greece)
Thales (640-546 B.C.)

SIMILAR TRIANGLES
Ex. 1. Any two line segments are similar.
2. Any two equilateral triangles are similar
3. Any two squares are similar.
4. Any two circles are similar.

We use the symbol '~' to indicate similarity of figures.

SIMILAR TRIANGLES

ΔABC and ΔDEF are said to be similar, if their corresponding angles are equal and the corresponding sides are proportional.

THEOREM-1 (Thales Theorem or Basic Proportionality Theorem)  : If a line is drawn parallel to one side of a triangle intersecting the other two sides, then the other two sides are divided in the same ratio.

Given : A ΔABC in which line  parallel to BC (DE  ||  BC) intersecting AB at D and AC at E.
To prove : AD /DB =AE EC
Construction : Join D to C and E to B. Through E drawn EF perpendicular to AB i.e., EF ⊥  AB and through D draw DG ⊥ AC.

Hence proved.

THEOREM-2 (Converse of Basic Proportionality Theorem) : If a line divides any two sides of a triangle proportionally, the line is parallel to the third side.
Given : A  ΔABC and DE is a line meeting AB and AC at D and E respectively such that AD/DB = AE/EC
To prove : DE || BC

Proof

Hence, proved.

Ex.1 In the adjoining figure, DE || BC.
(i) If AD = 3.4 cm, AB = 8.5 cm and AC = 13.5 cm, find AE. 
(ii) If AD/DB = 3/5 and AC = 9.6 cm, find AE.

Sol. (i) Since DE || BC, we have AD/ AE=AB/ AC   

Ex.2 In the adjoining figure, AD = 5.6 cm, AB = 8.4 cm, AE = 3.8 cm and AC = 5.7 cm. Show that DE || BC.

Sol. We have, AD = 5.6 cm, DB = (AB – AD) = (8.4 – 5.6) cm = 2.8 cm.
AE = 3.8 cm, EC = (AC – AE) = (5.7 – 3.8) cm = 1.9 cm.

DE divides AB and AC proportionally.
Hence, DE || BC

Ex.3 Prove that the internal bisector of an angle of a triangle divides the opposite side in the ratio of the sides containing the angle. (Internal Angle Bisector Theorem)

Sol. Given : AΔABC in which AD is the internal bisector of (angle)A
To Prove : BD/DC = AB/ AC
Construction : Draw CE || DA, meeting BA produced at E.
Proof :

Hence, Proved.

Remark : The external bisector of an angle of a triangle divides the opposite side externally in the ratio of the sides containing the angle. i.e., if in a ΔABC, AD is the bisector of the exterior of angle ∠A and intersect BC produced in D, BD/CD= AB/AC .