Linear Equations in Two Variables
Introduction to Euclid’S Geometry
Lines and Angles
- Concept of Quadrilaterals - Sides, Adjacent Sides, Opposite Sides, Angle, Adjacent Angles and Opposite Angles
- Angle Sum Property of a Quadrilateral
- Types of Quadrilaterals
- Another Condition for a Quadrilateral to Be a Parallelogram
- Theorem of Midpoints of Two Sides of a Triangle
- Property: The Opposite Sides of a Parallelogram Are of Equal Length.
- Theorem: A Diagonal of a Parallelogram Divides It into Two Congruent Triangles.
- Theorem : If Each Pair of Opposite Sides of a Quadrilateral is Equal, Then It is a Parallelogram.
- Property: The Opposite Angles of a Parallelogram Are of Equal Measure.
- Theorem: If in a Quadrilateral, Each Pair of Opposite Angles is Equal, Then It is a Parallelogram.
- Property: The diagonals of a parallelogram bisect each other. (at the point of their intersection)
- Theorem : If the Diagonals of a Quadrilateral Bisect Each Other, Then It is a Parallelogram
- Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles
- Angle Subtended by a Chord at a Point
- Perpendicular from the Centre to a Chord
- Circles Passing Through One, Two, Three Points
- Equal Chords and Their Distances from the Centre
- Angle Subtended by an Arc of a Circle
- Cyclic Quadrilateral
Areas - Heron’S Formula
Surface Areas and Volumes
Statistics and Probability
On the number line, distances from a fixed point are marked in equal units positively in one direction and negatively in the other. The point from which the distances are marked is called the origin.
The number line to represent the numbers by marking points on a line at equal distances. If one unit distance represents the number ‘1’, then 3 units distance represents the number ‘3’, ‘0’ being at the origin. The point in the positive direction at a distance r from the origin represents the number r where as the negative direction at a distance r from the origin represents the number -r. Fig.
The perpendicular lines are actually obtained as follows : Take two number lines, calling them X′X and Y′Y. Place X′X horizontal as following fig.
and the numbers on it just as written on the number line. We do the same thing with Y′Y except that Y′Y is vertical, not horizontal in following fig.
Combine both the lines in such a way that the two lines cross each other at their zeroes, or origins in following fig.
The horizontal line X′X is called the x - axis and the vertical line YY′ is called the y - axis. The point where X′X and Y′Y cross is called the origin, and is denoted by O. Since the positive numbers lie on the directions OX and OY, OX and OY are called the positive directions of the x - axis and the y - axis, respectively. Similarly, OX′ and OY′ are called the negative directions of the x - axis and the y - axis, respectively.
The axes (plural of the word ‘axis’) divide the plane into four parts. These four parts are called the quadrants (one fourth part), numbered I, II, III and IV anticlockwise from OX in following fig.
The Cartesian plane, or the coordinate plane, or the xy-plane. The axes are called the coordinate axes.
Shaalaa.com | Cartesian System
Write the answer of the following question:-
What is the name of horizontal and the vertical lines drawn to determine the position of any point in the Cartesian plane?
The point which lies on y-axis at a distance of 5 units in the negative direction of y-axis is ______.
Without plotting the points indicate the quadrant in which they will lie, if ordinate is 5 and abscissa is – 3
Without plotting the points indicate the quadrant in which they will lie, if abscissa is – 5 and ordinate is 3
Without plotting the points indicate the quadrant in which they will lie, if ordinate is 5 and abscissa is 3
In figure, LM is a line parallel to the y-axis at a distance of 3 units. What are the coordinates of the points P, R and Q?