Geometry is the branch of mathematics that deals with the properties, measurement and relations of points, lines, angles, surfaces and solids.
Geometry is such a vast topic and therefore it is almost an impossible endeavour to gather or compile all the concepts pertaining to this in a single book. Therefore, we will be discussing only the relevant concepts with respect to the type of questions that are posed in the CSIR-NET examination. With a proper understanding of these concepts, you will be able to attempt any geometrical questions in the CSIR-NET examination.
Geometry can be divided into plane geometry and solid geometry.
A plane is any two-dimensional surface such as circles and squares. Plane geometry is all about figures in a plane.
Let us discuss some important aspects in plane geometry.
Types of lines
- Intersecting lines
Two or more lines that meet at a point are called intersecting lines. The common point is called the point of intersection.
- Concurrent Lines
- Perpendicular Lines
A line is said to be perpendicular to another line if the two lines intersect at a right angle (90O angle).
intersect or touch at any point are called parallel lines.
Parallel lines and the transversal
A transversal is a line that intersects two or more other lines. When it intersects parallel lines, the following angles are congruent (equal).
Corresponding angles – Angles that are in the same position on each line.
Alternate interior angles – Angles on the opposite sides of the transversal and on the interior of the parallel lines.
Alternate exterior angles – Angles on opposite sides of the transversal and on the exterior of the parallel lines.
Vertically opposite angles – The pairs of opposite angles formed between each line and the transversal.
Note: Angles on one side of a straight line will always add to 180 degrees [Linear Pair].
Types of triangles
- Right triangles
A triangle with a right angle (That is, 90o) is called a right triangle.
In a right triangle,
(Hypotenuse)2 = (Base)2 + (Height)2
The numbers that satisfy this relation are called Pythagorean triplets.
Example: (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61),
(12, 35, 37), (15, 20, 25) etc.
Note: If you see a triangle with sides exactly as any of these Pythagorean triplets, you can be sure that the triangle is a right triangle.
- Isosceles triangles
Triangles with two equal sides are called isosceles triangles.
The angles opposite to the equal sides are also equal.
- Equilateral Triangles
Triangles with three equal sides are called equilateral triangles.
For equilateral triangles, each of the three angles are also equal to 60o.
Important facts about triangles
- Sum of the interior angles of a triangle is 180o (Angle Sum Property)
- The sum of any two sides of a triangle is greater than the third side.
- The diagonal of a square divides the square into two isosceles right triangles.
- Median – A median of a triangle is a line segment from one vertex to the mid-point on the opposite side of the triangle.
Centroid – A centroid of a triangle is the point where the three medians of the triangle meet.
- Orthocenter – In a triangle, the orthocenter is the point of intersection of the altitudes.
- Incircle – The incircle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is called the triangle’s incenter.
Circle is the set of all points in a plane that are at a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius. The diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle.
Chord is the line that joins two points on a circle. Diameter is the largest chord.
An arc of a circle is two points on the circle and the continuous part of the circle between the two points.
If the arc measures xo, the arc length is (2πr).
A tangent to a circle is a line that touches the circle at exactly one point, never entering the circle’s interior.
Important circle theorems
- A tangent and radius meet at right angles.
- Two tangents from a common exterior point to a circle are equal in length.
- The angle subtended by an arc at the center of a circle is twice the angle subtended by the same arc at the circumference of the circle.
- Angles subtended by an arc in the same segment of a circle are equal.
- The angle subtended in a semicircle is a right angle (90o).
Important properties of Parallelogram
- Opposite sides are parallel and equal
- Opposite angles are equal.
- The area of a parallelogram is twice the area of a triangle created by one of its diagonals.
- Diagonals of a parallelogram bisect each other. That is, they are divided in their intersection point into two.
- Parallelogram law – The sum of squares of diagonals is equal to the sum of squares of four sides.
A parallelogram with all sides equal is called a rhombus.
Important properties of Rhombus
- Opposite sides are parallel and all sides are equal.
- Opposite angles are equal.
- The diagonals of a rhombus are perpendicular to each other.
- Diagonals of a rhombus bisect each other.
- As Rhombus is a special parallelogram, it also follows the parallelogram law.
“For detailed theory, refer the book “CSIR-NET General Aptitude – A New Outlook”