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The coordinate proof is a proof of a geometric theorem which uses "generalized" points on the Cartesian Plane to make an argument. The method usually involves assigning variables to the coordinates of one or more points, and then using these variables in the midpoint or distance formulas.

Every place on this planet has coordinates that help us to locate it easily on the world map. The coordinate system of our earth is made up of imaginary lines called latitudes and longitudes. The zero degrees 'Greenwich Longitude' and the zero degrees 'Equator Latitude' are the starting lines of this coordinate system. Similarly locating the point in a plane or a piece of paper, we have the coordinate axes with the horizontal x-axis and the vertical y-axis.

Coordinate geometry is the study of geometric figures by plotting them in the coordinate axes. Figures such as straight lines, curves, circles, ellipse, hyperbola, polygons, can be easily drawn and presented to scale in the coordinate axes. Further coordinate geometry helps to work algebraically and study the properties of geometric figures with the help of the coordinate system.

Coordinate geometry is an important branch of math, which helps in presenting the geometric figures in a two-dimensional plane and to learn the properties of these figures. Here we shall try to know about the coordinate plane and the coordinates of a point, to gain an initial understanding of Coordinate geometry.

The distance between two points (x1, y1) and (x2, y2) is equal to the square root of the sum of the squares of the difference of the x coordinates and the y-coordinates of the two given points. The formula for the distance between two points is as follows.

D = √(x2−x1)^{2}+(y2−y1)^{2}

*All Formulas of Coordinate Geometry*

General Form of a Line: ax + by + c = 0

Slope Intercept Form of a Line: y = mx + c

Point-Slope Form: y − y1= m (x − x1)

The slope of a Line Using Coordinates: m = Δy/Δx = (y2 − y1)/ (x2 − x1)

The slope of a Line Using General Equation: m = −(A/B)

Intercept-Intercept Form: x/a + y/b = 1

Distance Formula: |P1P2| = √[(x2 − x1)2 + (y2 − y1)2]

For Parallel Lines: m1 = m2

For Perpendicular Lines: m1m2 = -1

** * **Midpoint Formula: M (x, y) = [½ (x1 + x2), ½ (y1 + y2)]

Angle Formula: tan θ = [(m1 – m2)/ 1 + m1m2]

Area of a Triangle Formula: ½ |x1(y2−y3) + x2(y3–y1) + x3(y1–y2) |

Distance from a Point to a Line: d = [|Ax0 + By0 + C| / √(A2 + B2)]

*Example 1*

Prove or disprove that the quadrilateral defined by the points (8,1), (6,9), (4,0), (2,8) is a rectangle.

It helps to start by plotting the points in the graph.

This shape * appears* to be a rectangle.

*Example 2*

Prove or disprove that the quadrilateral defined by the points (4,0), (5,3), (1,1), (2,4) is a square.

First, plot the points. The points have been labeled with letters to help with the identification of them later.

This shape * appears* to be a square. To

First: Find the lengths of all sides and verify that they are equal. One can use the distance formula or the Pythagorean Theorem to do this.

- AB = √3
^{2 }+ 1^{2}= √10 - BC = √3
^{2 }+ 1^{2}= √10 - CD = √3
^{2 }+ 1^{2}= √10 - DA = √3
^{2 }+ 1^{2}= √10

Because all four sides are the same length, all four sides are congruent.

Second, find the slopes of all four sides and verify that adjacent sides have opposite reciprocal slopes and therefore are perpendicular, creating right angles.

Slope of AB = −1/3

Slope of BC = 3

Slope of CD = −1/3

Slope of DA = 3

Because all adjacent sides have opposite reciprocal slopes, adjacent sides are perpendicular. This means that adjacent sides meet at right angles and the shape has four right angles.

Hence, it is a Square.

*Example 3*

Prove or disprove that the quadrilateral defined by the points (5, −1), (6,3), (1,1), (2,5) is a rhombus.

First, plot the points.

This shape * appears* to be a rhombus. To

- AB = √4
^{2}+2^{2}= √18 = 3√2 - BC= √4
^{2}+1^{2}= √17 - CD= √4
^{2}+2^{2}= √18 = √3 √2 - DA= √4
^{2}+1^{2 }= √17

Even though the shape * looked* like a rhombus, its four sides are not actually congruent. Therefore, this is NOT a rhombus.

Prove or disprove that the quadrilateral defined by the points (8,1), (6,9), (4,0), (2,8) is a rectangle.

It helps to start by plotting the points in the graph.

This shape * appears* to be a rectangle.

For Example:

A right triangle has legs of 5 units and 12 units. Place the triangle in a coordinate plane. Label the coordinates of the vertices and find the length of the hypotenuse.

One leg is vertical and the other leg is horizontal, which assures that the legs meet at right angles. Points on the same vertical segment have the same x-coordinate, and points on the same horizontal segment have the same y-coordinate.

We can use the Distance Formula to find the length of the hypotenuse.

Distance Formula:

d = √[(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}]

Substitute (x_{1}, y_{1}) = (0, 0), (x_{2}, y_{2}) = (12, 5).

d = √[(12 - 0)^{2} + (5 - 0)^{2}]

Simplify.

d = √(12^{2} + 5^{2})

d = √(144 + 25)

d = √169

d = 13

The Method of Coordinates is a way of transferring geometric images into formulas, a method for describing pictures by numbers and letters denoting constants and variables. It is fundamental to the study of calculus and other mathematical topics.

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