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This essential trains for: SAT-I, SAT-II, GMAT, AMC-8, AMC-10.

Coordinate geometry requires finding distances, angles, areas, etc. for figures that are specified through coordinates in a Cartesian system of coordinates - 2D or 3D.

The coordinates of a point are the distances from the point to the axes of coordinates. Since the system is Cartesian, the distances are perpendicular segments.

The distance from the point to the x-axis is called an x-coordinate.

The distance from the point to the y-axis is called an y-coordinate.

In 3D, the distance from the point to the z-axis is called an z-coordinate.

Coordinates are specified in a list ordered as follows: (x, y, z).

figure

The coordinates of the midpoint of a segment AB with endpoints A=(xA, yA) and B=(xB, yB) are:

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figure

The distance between two points in the coordinate plane is easily found using the Pythagorean theorem:

figure

The horizontal distance between points A and B is:

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The vertical distance between points A and B is:

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Note: the absolute value is there because the distance is always a positive number.

The distance between points A and B represents the hypotenuse of a right triangle with legs 12 and 5:

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Since the square makes the numbers positive anyway, we can write the general formula for the distance between two points in 2D Cartesian coordinates as:

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In 3D, the formula generalizes as:

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The area of a scalene triangle given in coordinates can be easily found by inscribing it in a rectangle and subtracting three right triangles:

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The rectangle has area:

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The slope of a line is "rise over run," where 1 and 2 designate two points arbitrarily chosen on the line:

equation

figure

Parallel lines have the same slope.

Perpendicular lines have slopes that multiply up to -1:

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Data sufficiency: There are an infinity of parallel lines - therefore, just knowing the slope does not identify the parallel line uniquely. To completely determine a line that is parallel to a given one, one has to know one additional fact such as: the y-intercept, the coordinates of a point that belongs to the line, the x-intercept, etc.

Same for perpendicular lines.