Coordinate Geometry - GMAT Math Study Guide

Cartesian Coordinate System

The Cartesian coordinate system, shown below, consists of an X-axis (which runs horizontally) and a Y-axis (which runs vertically). The Cartesian coordinate system, also known as the coordinate plane, is used to graph lines, circles, parabolas, points, and other mathematical objects.

The power of the coordinate plane lies in the use of ordered pairs. The ordered pair (5,-2) refers to the point which has an x value of 5 and a y value of -2. Stated differently, the pairing is in the form (x, y). If two or more points are connected, a line or curve is formed.

The following terms are used when interacting with coordinate planes:

• X-Axis - The horizontal line running through the center of the graph from left to right.
• Y-Axis - The vertical line running through the center of the graph from bottom to top.
• Ordered Pair - The means of identifying a point through its coordinates. The proper notation is: (X, Y) where (0, 0) is the intersection of the x and y-axis.
  For example, point A is at (2, 4) since it is horizontally 2 units to the right of the center and it is vertically 4 units above the center.
  Similarly, point C is at (-6, -2) since it is horizontally 6 units to the left of the center and it is vertically 2 units below the center.
  Point E: (0, 0)
  Point B: (-7 , 5)
  Point D: (2, -1)
• Origin - The point in the center of the coordinate plane where the x and y axis intersect (0, 0). The origin is point E in this graph.

Quadrants

Each coordinate plane is divided up into four quadrants, labeled below. (Note: Some graphs only show one quadrant. In this case, the other quadrants still exist, but they are merely not shown).

In the first quadrant, both x and y are positive while in the second quadrant x is negative and y is positive. The chart below depicts the sign of x and y [denoted (X, Y)].

Quadrant I: (+, +)
Quadrant II: (-, +)
Quadrant III: (-, -)
Quadrant IV: (+, -)

Slope

One property of a line is its slope, which is a measure of the steepness of the line. Every line has a slope defined by rise over run (i.e., the amount the line rises vertically over the amount the line runs horizontally). Rise over run refers the change of the rise (y values) of any two points on the line over the change in the run (x values) of the same two points on the line. For example:

In the above graph, point A is at (-8, 4) and point B is at (8, -4).

Slope Categories

There are four types of slope: positive, negative, zero, and undefined.

• Blue Line - Positive Slope
• Red Line - Negative Slope
• Green Line - Slope of Zero
• Brown Line - Undefined Slope

Lines

y = mx+b

In the above coordinate planes, the lines appeared without any explanation as to why the line pointed in a certain direction at a certain steepness. The location and slant of a line is determined by an equation. It is the line that graphs out all the points that satisfy this equation. Since this is important, it bears repeating: a line on a coordinate plane is a graphical representation of a series of points that fulfill a mathematical equation.

The standard form in which linear equations which are graphed appear on a coordinate plane is:

Consider the following example:

Graph the series of points satisfying the equation y = 2x + 3

One means to do this would be to manually generate a list of points that satisfy this equation. For example, if x = 0, y must equal 3; if x = 1, y must equal 5; etc.
However, there is a faster way. According to the y = mx + b formation of a line, m = 2 and b = 3. Consequently, the line being graphed must cross the vertical axis 3 units above the horizontal axis and it must rise vertically 2 units for every 1 unit it runs horizontally.

Horizontal and Vertical Lines

• A horizontal line can be written as y = b since for each value on the line, the y-coordinate will be the same (regardless of the x-coordinate). Since the line does not rise when it runs, the slope, m, is 0.
  In the coordinate plane above, the light blue line can be written as: y = 3
• A vertical line can be written as x = n since for each value on the line, the x-coordinate will be the same (regardless of the y-coordinate). However, since the line does not run when it rises, the slope is ^∞/₀, which is undefined since you cannot divide by zero.
  In the coordinate plane above, the red line can be written as: x = 3

Writing the Equation of a Line

It is important to know how to take a pair of points (whether from a graph or from a word problem) and write an equation for a line that satisfies the two points. This process involves solving for m and b in the equation y = mx + b. Consider the following example:

Axis Intercepts

The x and y-intercept are important properties of a line and it is often necessary to find the exact location where a line intersects the x-axis and y-axis. The best means to find an intercept is algebraically.

X-Axis Intercept

When a line crosses the x-axis, its y-value will be zero. Consequently, by setting y = 0 and solving for x, the x-coordinate at which the line crosses the x-axis can be found.

Y-Axis Intercept

When a line crosses the y-axis, its x-value will be zero. As a result, by setting x = 0 and solving for y, the y-coordinate at which the line crosses the y-axis can be found. If an equation is in y = mx+b format, recall that since setting x = 0 yields y = b, the value of b is the y intercept.

Parallel Lines

Parallel lines are lines that never intersect. In order to never intersect, two lines must have the same angle (technically called slope). If two lines do not have the same slope, they will eventually intersect. However, if two distinct lines have the same slope, they will never intersect.

Perpendicular Lines

Line A is perpendicular to Line B if Line A intersects Line B at a 90° angle. The most important property of perpendicular lines is as follows:

Distance Between Points

In order to find the distance between two points, either: (1) use the distance formula [to be derived] or (2) draw a triangle and use the Pythagorean theorem.

The distance formula comes from the Pythagorean theorem, as the example below shows:

In the graph below, what is the distance between points K and L?

1. The best means to solve this type of a question is by drawing in a triangle and solving for the hypotenuse, which is the distance between points K and L. In order to do this, sketch in a triangle by placing a third point such that a right angle is formed (point N below is such a point):
   
2. By inspection, the location of each point is as follows:
   L: (2, 1)
   N: (5, 1)
   K: (5, 5)
3. Find the length of each leg of the right triangle:
   LN = 5 - 2 = 3
   KN = 5 - 1 = 4
4. Use the Pythagorean theorem to find the length of KL:
   (KL)² = (LN)² + (KN)² =
   (KL)² = 3² + 4²
   (KL)² = 9 + 16
   (KL)² = 25
   KL = 5

Distance Formula

The process above can be simplified using the following formula:

The coordinates of K are x₁, y₁ or (5, 5).
The coordinates of L are x₂, y₂ or (2, 1).
d = KL

Notice that the distance formula immediately above is simply a formulaic representation of the graphical process undertaken above to solve for KL.

Types of GMAT Problems

1. Finding the Equation of a Line Given Two Points

   Given two points on a line, it is possible to find the equation of that line. Begin by calculating the slope (i.e., the rise over run). Plug the slope in for m in y = mx + b. Solve for b by plugging in one of the ordered pairs for x and y. Finally, substitute the answer for b back into the equation.

   Which of the following is the equation of a line that goes through the point (10,5) and has an x-intercept of 5.

   A)	y = x - 4
   B)	y = x - 5
   C)	y = 2x - 4
   D)	y = 2x - 5
   E)	y = 3x - 4

   Correct Answer: B

   1. If a line has an x-intercept of 5, by definition, the line must go through the point (5,0). You now have two points, which will form a line: (5,0) and (10,5).
   2. Calculate the slope of the line (i.e., rise over run).
      Rise = change in y = 5 - 0 = 5
      Run = change in x = 10 - 5 = 5
      Slope = Rise/Run = 5/5 = 1
   3. Substitute the slope, 1, in for m.
      y = mx + b
      y = 1x + b
      y = x + b
   4. Substitute an ordered pair in for x and y and solve for b, which is the intercept. Use (10, 5).
      5 = 1*10 + b
      5 -10 = b
      b = -5
   5. Note: You could have substituted in the other point too. Use (5, 0).
      0 = 1*5 + b
      0 -5 = b
      b = -5
   6. Substitute b=-5 into the equation y = x + b
      y = x - 5
2. Finding the Distance Between Two Points

   The Pythagorean Theorem can be used to find the distance between two points. Recall that the Pythagorean Theorem stated, a²+ b² = c². Think of a as representing the difference between the x values and b representing the difference between the y values. Then, think of c as the distance between the two points.

   What is the shortest distance between points (-2, 1) and (2,-2)?

   A)	1
   B)	2
   C)	3
   D)	4
   E)	5

   Correct Answer: E

   1. The shortest distance between any two points is a line.
   2. Points are written in the (x,y) format, where x is the x-coordinate and y is the y-coordinate.
   3. Calculate the difference between the x and y values of each coordinate.
      Difference between y values = 1 - (-2) = 1 + 2 = 3
      Difference between x values = (-2) - 2 = -4
   4. If you start at (2, -2) and travel over -4 units (i.e., travel to the left) and then travel straight up 3 units, you will end up at (-2, 1). This sketches out a right triangle, where the hypotenuse is the length of the shortest distance between the two points (i.e., a straight line between the two points).
   5. Use the Pythagorean Theorem with a = -4 and b = 3 to solve for this distance between the two points.
      a²+ b² = c²
      (-4)²+ 3² = c²
      
   6. Solve for c
      16 + 9 = c²
      25 = c²
      c = 5 (Negative 5 cannot be a solution to a length so it is discarded)
3. Problems Involving Quadrants

   The axes of the coordinate plane separate the space into four quadrants. The first quadrant is where both x and y are positive. The second quadrant has positive y and negative x values. The third quadrant has negative x and y values. Finally, the fourth quadrant has positive x and negative y values.
   To determine what quadrant a line goes through, find the x and y intercepts. Then, draw a rough graph of the line.

   What quadrants do the line y = 2x - 4 go through?

   A)	1 and 2
   B)	1 and 3
   C)	1 and 4
   D)	1, 2 and 3
   E)	1, 3 and 4

   Correct Answer: E

   1. Find the x-intercept and the y-intercept of the line.
      To find the x intercept, set y = 0 and solve for x.
      0 = 2x - 4
      x = 2
      
      To find the y intercept, set x = 0 and solve for y.
      y = 0 - 4
      y = -4
   2. Draw the line using the two intercepts.
      
   3. Thus, the line goes through the 1st, 3rd and 4th quadrants.