Table of Contents
Definitions
Parallel lines are lines that never intersect, no matter how far they are extended. Parallel lines have the same slope but different x and y intercepts.
Perpendicular lines are lines that intersect at a 90 degree angle. Perpendicular lines have an inverse slope with each other.
Intersecting Lines
When two lines intersect the supplemental angles sum to 180 degrees
Angles a and b sum to 180. Angles c and d sum to 180. angle a = angle d, angle b = angle c. Together angles a, b, c, d sum to 360 degrees.
X, Y Coordinates
The coordinate plan is formed by two axes, x and y. The x axis is the horizontal axis, and the y axis is the vertical axis. The position of a given point is often provided as an ordered pair using the format (x,y)
The slope can be found by determining the change in y over the change in x : Δy/Δx
Example
The slope of the line above is -3/7. Notice how the slope is negative because from point to point the change in y is negative 3 and the change x is positive 7.
The point where a line intersects with the coordinate axis is called the intercept. Lines can have both an x and y intercepts. Intercepts can be easily found by setting one of the coordinate variables equal to zero.
Using the figure above, the y intercept (the y value when x is zero) is (1,0)
The x intercept (or the x value when y is zero) is (3,0)
Line Equation
Lines are often provided on the GMAT in the form of:
y = mx + b
where m is the slope and b is the y-intercept
Once an equation is known, it can be manipulated to find a wealth of information.
Distance Between Two Points
The distance between two points can be found by applying a rule we have already reviewed: the pythagorem theorem!
Using the points given above, imagine the two connected points form a right triangle. We can determine:
a² + b² = c²
3² + 4² = c²
25 = c²
c = 5
The distance between the two points is equal to 5
Perpendicular Bisector
This is a problem type that frequently arises in some form on the GMAT.
Example
In the diagram, the line y = 4 is the perpendicular bisector of segment JK (not shown). What is the distance from the origin to point K?
The perpendicular bisector intersects line JK right through the midpoint. Since the distance from line y to point J is 6, this means that point K has coordinates (6,-2).
Plug the coordinates of point K into the distance equation:
c² = 6² + 2²
c² = 36 + 4
c² = 40
c = 2√10
The distance from point K to the origin is 2√10
Quadrants
Quadrants are a very common test topic in data sufficiency questions. Memorize the following quadrant order. This is a standard used in geometry.
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