1

Lines & Triangles

1. Introduction

Euclid, a mathematician who lived about 2300 years ago, wrote the Elements, a book which included definitions, properties and proofs for plane geometry wherein two points on a plane can always be connected by a straight line. This is the Geometry we learnt at school and is a part of the curriculum for MBA entrance tests. Other forms of Geometry like Elliptic, Hyperbolic, etc. are not part of this course.

Euclidean Geometry is the part of Mathematics that deals with figures, shapes and sizes, of 0, 1 and 2 dimensions. One has to have a keen sense of visualisation combined with a thorough knowledge of theorems to work fluently in Geometry.

In this lesson, we will start with the properties of points and lines before moving on to Triangles. Properties pertaining to polygons and circles are covered in the subsequent lessons. Properties pertaining to 3-dimensional shapes are detailed in the Mensuration Lesson.

1.1 Revisiting Basics

Let's start with the recap of the basics of Geometry, learnt at school.

Zero-Dimensional

A point does not have any dimension (like length, breadth or height).It is the basic unit of any shape. Shapes are a collection of points.

Though one might not be questioned exclusively on points, it is imperative for you to understand the following.

Collinear Points are points which lie on the same line.

Vertex is a point where 2 or more lines or curves meet.


In the figure above, A, B and C are the vertices


In the figure above, P, Q, R, S, T and U are the vertices

Example 1:

If AB = 13 cm, BC = 8 cm and points A, B and C are collinear, then AC = ?

(1) 13 cm (2) 21 cm (3) 13 cm or 21 cm (4) 5 cm or 21 cm


Solution:

AB is the longer side of 13 cm length. There are two possibilities here.

Case 1: If B is between A and C, then

AC = AB + BC = 13 + 8
$\implies$ AC = 21cm

Case 2: C is between A and B

AC + CB = AB
$\implies$ AC = AB - CB = 13 - 8
$\implies$ AC = 5 cm

Answer: (4) 5 cm or 21 cm