Geometry Basics

The SBC GMAT Files

GMAT Primer: Plane And Coordinate Geometry Basics

Haven’t done Geometry since high school? No worries! Here are the basic formulas and definitions you need to know to get started. Create a “Geometry Formulas” cheat-sheet to write down these formulas and refer back to it as needed when you study!

An angle is formed by two lines or line segments which intersect at one point. The point of intersection is called the vertex. Angles are measured in either degrees or radians.

An acute angle is an angle whose measurement in degrees is between 0 and 90. A right angle is an angle whose measurement in degrees is exactly 90. An obtuse angle is an angle whose degree measure is between 90 and 180. A straight angle is an angle whose degree measure is exactly 180 degrees.

All of the angles on one side of a straight line sum to 180 degrees.

a + b + c = 180 degrees

All of the angles around one point must sum to 360 degrees.

a + b + c + d + e = 360 degrees

Perpendicular lines are formed when the angle between two lines is 90 degrees. The shortest distance from a point to a line is a line with a length such that the two lines form a 90 degree angle.

Two angles are supplementary if they share one line; i.e., if they sum of their angles is 180 degrees. Two angles are complementary if together they make a right angle; i.e., if the sum of their angles is 90 degrees.

To bisect an angle means to cut it in half. The two smaller angles will then have the same measurement.

If two parallel lines intersect with a third line, the third line is called a transversal. When this happens, all acute angles are equal and all obtuse angles are equal. Each acute angle is supplemental to each obtuse angle.  Vertical angles are a pair of opposite angles formed by intersecting lines, and they are equal.

There are two main equations for straight lines. One form looks like:

For an equation that looks like this the slope is and the y intercept is .For example, in the equation 2x + 3y + 6 = 0, the slope is -2/3 and the y-intercept is -2.
The second equation is called slope-intercept form and looks like:  .  Here m is the slope and b is the y-intercept.
Distance Formula       Use this to find the distance between two points.

Midpoint Formula

Use this to find the midpoint between two points (notice how you are essentially finding the average of the x-coordinates and the average of the y-coordinates).

Slope = Rise / Run = Change in y / Change in x

The standard equation for a parabola is y = ax2 + bx + c. In this equation c represents the y-intercept. A standard equation in which a variable is squared will never make a straight line.

The x-intercepts are also called the “roots” or “solutions” of a parabola. On the GRE, parabolas will often be referred to as “functions” interchangeably. The x-intercepts can be found using the quadratic formula:

To find the number of x-intercepts a given parabola has, calculate what is called the discriminant: , or the information underneath the radical in the quadatic formula.

If the discriminant is positive, the parabola has two intercepts with x-axis; if it is negative there are no intercepts with the x-axis, and if the discriminant is equal to zero there is one intercept with the x-axis.

The vertex represents the maximum (or minimum) value of the function. Think of it as the starting point of the function.
The vertex of the parabola is located at point  for the standard equation. If you are given the standard equation, you can find the vertex and the x-intercepts.

The standard equation of a circle is (x – h)2 + (y – k)2 = r2   where (h, k) is the center point of the circle and r is the radius.

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