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28 January 2021
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Geometry! Geometry everywhere!
Coordinate Geometry involves the use of algebraic processes in the study of geometric problems and also the geometric interpretation of algebraic equations.
It is important that the reader be able to not only use algebraic processes but also to understand them fully if he is to apply them with confidence.
What is Coordinate Geometry?
A system of geometry where the position of points on the plane is described using an ordered pair of numbers.
It is socalled coordinate geometry because it uses the treatment of geometric problems a system of coordinates, which associates with each point of a geometric figure, a set of numbers coordinates so that the conditions which each point must satisfy can be expressed in terms of equations or inequalities ordinarily involving algebraic quantities and at times trigonometric functions.
It is also called Analytical Geometry as it describes the link between algebra and geometry, where there is the use of curves and lines on graphs.
Introduction to Coordinate GeometryPDF
Coordinate Geometry involves the use of algebraic processes in the study of geometric problems and also the geometric interpretation of algebraic equations. Here is a downloadable PDF to explore more.
ğŸ“¥  Introduction to Coordinate GeometryPDF 
Let’s understand Coordinates
Here is the figure which shows the concept of Coordinate Geometry.

Let’s see the grid above. Digits 1, 2, 3, 4, 5 etc. are marked on the columns of the grid, the rows have letters as A, B, C, and so on.

Observe letter X is situated in the square 3B (column 3 and row B.) So, the coordinates of this square are 3 and B.

When we observe closely, in the grid there are mainly two sections: Row and Column. So there might be many such rows and columns.

The point where the row and column intersect will be our square marked.
Important Terminologies of Coordinate Geometry
As a basis for the definition of coordinates, we take two lines perpendicular to one another, which are called the xaxis (X) and yaxis (Y) respectively; their intersection is called the origin (O).

ORIGIN: The point where the axes cross, where both x and y are zero.
 ABSCISSA: xcoordinate of a point. The distance along the horizontal axis.

ORDINATE: ycoordinate of a point. The distance from the xaxis is measured parallel to the yaxis.
 QUADRANT: The coordinate axes divide the plane into four compartments, which are called quadrants.

Quadrant 1 : (+x, +y): The quadrant formed by the positive xaxis and positive yaxis is called the first quadrant;

Quadrant 2 : (x, +y): The one to the left of the positive yaxis (and above the xaxis), is the second quadrant;

Quadrant 3 : (x, y): The ones below the negative xaxis, is called the third quadrant.

Quadrant 4 : (+x, y): The ones below the positive xaxis, is the fourth quadrant.
Important Formulas of Coordinate Geometry

Distance Formula
Distance between Two Points on the Same Coordinate Axes
The distance between two points which are on the same axis (xaxis or yaxis), is given by the difference between their ordinates if they are on the yaxis, else by the difference between their abscissa if they are on the xaxis.
Distance‘d’ between any two points (x_{1}, y_{1}) and (x_{2}, y_{2}) is given by
\[d=\sqrt{[(x_2x_1)^2+(y_2y_1)^2]}\]
Example 
a) Find length of \(\bar{AB.}\)
Length of AB
\[\begin{align}A B&=\sqrt{\left[\left(X_{2}X_{1}\right)^{2}+\left(y_{2}y_{1}\right)^{2}\right]} \\[6pt]A B&=\sqrt{\left[(41)^{2}+(2\{3\})^{2}\right]} \\[6pt]A B&=\sqrt{\left[(41)^{2}+(2+3)^{2}\right]} \\[6pt]A B&=\sqrt{\left[(3)^{2}+(5)^{2}\right]} \\[6pt]A B&=\sqrt{[9+25]} \\[6pt]A B&=\sqrt{[34]}\end{align}\]

Section Formula
If the point P(x, y) divides the line segment joining A(x_{1}, y_{1}) and B(x_{2}, y_{2}) internally in the ratio m:n, then, the coordinates of P are given by the section formula as:
\[\boxed{P(x, y) = \left(\frac{(mx_2+nx_1)}{m+n,} \frac{(my_2+ny_1)}{m+n}\right)}\]
This formula is useful to determine the coordinates of point P on line segment AB.
Let's say we have a line segment AB
Coordinates of A =(x_{1}, y_{1}) and B = (x_{2}, y_{2})
And a point is marked on the line
The ratio in which this point is marked is m:n
I.e. AP = m and PB = n
Applying the above section formula we can find coordinates of P

Midpoint
The midpoint of any line segment divides it in the ratio 1: 1. The coordinates of the midpoint (P) of the line segment joining A(x_{1}, y_{1}) and B(x_{2}, y_{2}) is given by
\[P(x, y) = \left(\frac{(x_1+x_2)}{2}, \frac{(y_1+y_2)}{2}\right)\]
Taking above section formula as a base, when point P is marked exactly on center of a line segment AB where Coordinates of A =(x1, y1) and B = (x2, y2)
The ratio in which this point is marked is m: n
I.e. AP = m and PB = n
As it is exactly in center by f line segment the ratio is 1:1
Example 
\[\begin{align} \text { midpoint } &=\left(\frac{x_{2}+x_{1}}{2}, \frac{y_{2}+y_{1}}{2}\right) \\[6pt] &=\left(\frac{4+1}{2}, \frac{2+(3)}{2}\right) \\[6pt] &=\left(\frac{5}{2},\frac{1}{2}\right) \end{align}\]

Centroid Of A Triangle
If A(x_{1}, y_{1}), B(x_{2}, y_{2}) and C(x_{3}, y_{3}) are the vertices of a ΔABC, then the coordinates of its centroid (P) is given by
\[P(x, y) = \left(\frac{(x_1+x_2+x_3)}{3},\frac{(y_1+y_2+y_3)}{3}\right)\]

Area Of A Triangle Given By Its Vertices
If A(x_{1}, y_{1}), B(x_{2}, y_{2}) and C(x_{3}, y_{3}) are the vertices of a Δ ABC, then its area is given by
\[A = ½ [x_1(y_2 − y_3) + x_2(y_3 − y_1) + x_3(y_1 − y_2)]\]
Where A is the area of the Δ ABC.
This is one of the ways to find the area of a triangle using Coordinate Geometry.

Slope Formula
Example 
Line r passes through (2, 2) and (5, 8). Lines passes through (8, 7) and (2, 0). Is r ⊥ s?
See whether the slope of line r, m_{r} , and the slope of lines, m_{s} , are opposite reciprocals
Notice that \(m_r=\frac{6}{7}\) and \(m_s=\frac{7}{6}\) are opposite reciprocals. Therefore, the two lines are perpendicular.
Example 
An equation for line v is \(y = \frac{3}{2}x.\) An equation for line w is 6x + 4y = 7. Is v  w?
Both lines are parallel because the slope of line w and the slope of line v are the same. Both slopes are \(\frac{3}{2}.\)
Learning Outcomes
 Plotting points in the graph.
 Solving problems using formulas.
Example Problems
Question 1: (2, 1) is a point, which belongs to the line ____.
(A) x = y  (B) y=x+1  (C) 2y=x  (D) xy=1 
Question 2: The coordinates of two points are A (3, 4) and B (−2, 5), then (abscissa of A)  (abscissa of B) is ____.
(A) 1  (B) 1  (C) 5  (D) 5 
Question 3: Study the graph and answer the following question.
 The coordinates of point S are ____.
 Sum of abscissa of point P and R is ____.
 The point whose abscissa is 2 more than the ordinate is ____.
Question 4: Find the area (in square units) of the triangle whose vertices are (a, b+c), (a,b−c) and (−a, c).
(A) 2ac  (B) 2bc  (C) b(a+c)  (D) c(a−b) 
Question 5: Students of a school are standing in rows and columns in their playground for drill practice. A, B, C and D are the position of the four students as shown in the figure.
If Mohit wants to stand in such a way that he is equidistant from each of the four students A, B, C and D then what are the coordinates of his position?
(A) (6,6)  (B) (7,5)  (C) (6,4)  (D) (4,6) 
Question 6: To raise social awareness about the hazards of smoking, a school decided to start the "No smoking campaign". A student is asked to prepare a campaign banner in the shape of a triangle shown in the figure. If the cost of 1 cm^{2} of the banner is â‚¹2, find the cost of the banner.
(A) 12  (B) 6  (C) 5  (D) 10 
Question 7: If the centroid of the triangle formed by the points (a,b), (b,c) and (c,a) is the origin, then a^{3}+b^{3}+c^{3}=
(A) abc  (B) 0  (C) a+b+c  (D) 3abc 
Solutions
Q1: option B
Q2: option C
Q3: a) (5, 4 ) b) 3 c) Q
Q4: option A
Q5: option B
Q6: option D
Q7: option D
Summary
 Coordinates serve as the bridge between algebra and geometry. It is a bridge with twoway traffic; also by means of coordinates, geometric problems may be given its algebraic form.
 By this means, a geometric problem is reduced to its algebraic form, which most people can handle with greater ease and confidence.
 After the algebraic solution has been obtained, however, there remains its geometric interpretation to be determined; forgetting the problem is geometric, and algebra is a means to its solution, not the ultimate end.
 This use of coordinates was Descartes's great contribution to mathematics, which revolutionized the study of geometry.
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Frequently Asked Questions (FAQs)on Coordinate Geometry
Who discovered Coordinate Geometry?
This method was introduced by Rene Descartes in La Geometric, published in 1636; accordingly coordinate geometry is sometimes called Cartesian geometry.
Where do we use coordinate Geometry?
Below are Applications of Coordinate Geometry.
 Geometry deals with spatial concepts.
 Changing any shape or adding different colours.
 Uses in fields of trigonometry, calculus, dimensional geometry and more….
 Scanner and photo copying machine.
 The problems of physics, astronomy, engineering, etc. involve not only space but usually time also.
 The method of attack upon these problems is similar to that used in coordinate geometry.
Written by Dhanshree KB