Co-ordinates and scales
Introduction
The coordinates of two numbers or the Cartesian co-ordinates are located at a specific point on a grid known as the coordinate plane. The co-ordinates can be two numbers, or a number and a letter.
Types of Quadrants
A co-ordinate plane consists of $$4$$ quadrants and two axes.
In the first quadrant, the $$x$$-axis and $$y$$-axis values are both positive, such as $$(2,5)$$.
In the second quadrant, the value of the $$x$$-axis is negative, and the $$y$$-axis is positive, such as $$(-2,5)$$.
In the third quadrant, the values of the $$x$$-axis and $$y$$-axis are both negative, such as $$(-2,-5)$$.
In the fourth quadrant, the $$x$$-axis value is positive, and the $$y$$-axis is negative, such as $$(2,-5)$$.
Origin Point: The point that two axes interest each other is the origin point, $$(0,0)$$.
Scale: A scale refers to how the map units are related to the real-world units. Scale is represented by a fraction, a graphic scale, and a verbal description. The representative fraction is used for the scale, where the scale is shown in the ratio. In this, the numerator is $$1$$, and the denominator represents the greater distance.
E3.1: Demonstrate familiarity with Cartesian co-ordinates in two dimensions.
Two-dimensional Cartesian coordinates are defined by an ordered pair of vertical axes, the unit of measurement for each axis, and the path of each axis. The point at all the axis gets meet together is called the origin point for both. Suppose a point $$A$$ and the line drawn through $$A$$ is perpendicular to each axis. The position where the perpendicular cut the axis is explained as a number.
The two numbers that are chosen are the Cartesian coordinates of $$A$$.
The first co-ordinate is abscissa.
The second co-ordinate is known as the ordinate of $$A$$.
The points where axes get to meet are the origin of the co-ordinate system.
The representation of the co-ordinates is $$(x,y)$$, two numbers, written in the parenthesis separated by a comma.
The origin co-ordinates are $$(0,0)$$, whereas the coordinates on the positive-half sides one unit away are represented by $$(0,1)$$ and $$(1,0)$$. In this, the Euclidean plane with the Cartesian coordinate system is known as the Cartesian plane.
Cartesian plane: It is defined with the help of canonical representatives of few geometric figures.
Quadrant: The $$2$$ axes divide the plane into $$4$$ right angles of $$90^{\circ}$$, known as quadrants. And the quadrant where all the coordinates are positive is known as the first quadrant.
A real-life example is architectural engineers. While designing a house, they plan the building, and according to that, they build it.
Worked examples
Example 1: Richa and Rubi want to make a garden using co-ordinates $$(1,1)$$, $$(1,2)$$, $$(3,2)$$, and $$(3,1)$$. Richa says the ground will be a square based on the co-ordinates, whereas Rubi says it will be a rectangle. Identify who is right?
Step 1: Draw the coordinate on the cartesian plane.
Step 2: Identify the figure obtained.
The figure obtained with the help of the graph is a rectangle whose two parallel sides are equal and $$4$$ sides at an interior angle of $$90^{\circ}$$.
Rubi was right, as she has said that the ground would be a rectangle.
Example 2: Plot the points given below on the Cartesian coordinate system. Label the quadrant to which they belong.
$$(4,3)$$
$$(3,-4)$$
$$(-3,2)$$
$$(-2,-3)$$
Step 1: Draw the coordinate on the Cartesian plane.
Step 2: Explain which coordinate belongs to which quadrant.
Co-ordinates $$(4,3)$$ belong to the first quadrant, as the coordinate of the $$x$$-axis is positive, and the c-oordinate of the $$y$$-axis is positive.
Co-ordinates $$(3,-4)$$ belong to the fourth quadrant, as the coordinate of the $$x$$-axis is positive, and the co-ordinate of the $$y$$-axis is negative.
Co-ordinates $$(-3,2)$$ belong to the fourth quadrant, as the coordinate of the $$x$$-axis is negative, and the co-ordinate of the $$y$$-axis is positive.
Co-ordinates $$(-2,-3)$$ belong to the fourth quadrant, as the coordinate of the $$x$$-axis is negative, and the co-ordinate of the $$y$$-axis is negative.
In Cartesian co-ordinates, while plotting the coordinates in the Cartesian plane, always take care of the sign of the coordinate as it is negative or positive, and plot carefully in which quadrant they belong.