Page 3 Quantitative Reasoning Study Guide for the GRE

Geometry Skills

The xy Plane

The xy plane, also known as the Cartesian or coordinate plane, is a plane composed of 2 perpendicular axes that both extend to infinity. Points, lines, and other functions can be graphed on the coordinate plane. The intersection of the horizontal x-axis and the vertical y-axis creates four quadrants. Quadrant I contains positive x and positive y values, Quadrant II contains negative x values and positive y values, Quadrant III contains negative x and y values, and Quadrant IV contains positive x values and negative y values.

An ordered pair designates an x and y location for a point, for example: $(2, 3)$ designates a point located in Quadrant I right 2 and up 3.

The point(s) at which a line or function crosses the x-axis is called the x-intercept. The point at which a line or function crosses the y-axis is called the y-intercept.

A line is a function that represents a constant rate. The slope of a line is defined as: $m = \frac{y_2 - y_1}{x_2 - x_1}$ where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.

Equations describing a line are:

Slope-intercept form: $y = mx + b$, where m is the slope, and b is the y-intercept

Point-slope form: $(y - y_1) = m(x - x_1)$, where m is the slope and $(x_1, y_1)$ is a point on the line

Standard form: $Ax + By + C = 0$, where A, B, and C are real numbers

Graphing Functions

The xy coordinate plane enables graphing of functions in x and y variables. Graphing can be accomplished in many ways. An xy table, in which differing x values are substituted and evaluated in the function, will always produce a graph; however the process is tedious and it is difficult to ensure that all of the function’s behaviors are shown.

Familiarity with the different types of functions, and their general shape, reduces the effort necessary to graph them. For example:

$x = 5$ and $f(x) = 2$ are straight lines

$f(x) = mx + b$ is another straight line, with a slope of m and a y-intercept of b

$f(x) = \mid x+3 \mid$ produces two straight lines that have a vertex at a single point

$f(x) = ax^2 + bx + c$ produces a parabola with a line of symmetry at $x = \frac{-b}{2a}$ and up to 2 distinct roots

$f(x) = \sqrt{x}$ produces curved line above the x-axis and to the right of the y-axis starting at (0,0) and undefined when x is less than 0

$f(x) = a^x$ produces an exponential graph that increases in rate of change as x increases

$f(x) = x^3$ produces a parabolic figure extending in opposite directions

In cases where the x variable is combined with a positive or negative constant, the entire graph is shifted left or right the constant number of spaces.

$f(x) = (x - 3)^2$ is the general function $f(x) = x^2$ shifted 3 places to the right.

Additions or subtractions to the end of the function produce a vertical shift up or down for positive or negative values.

$y = 2x + 2$ is the general function $y = x$ with an increased slope and shifted 2 places upward.

Lines and Points

A point is a zero-dimensional that gives rise to lines and figures.

A line is a collection of points extending to infinity in both directions.

Intersecting lines give rise to a plane, a 2-D space that enables graphing of functions in x and y variables.

A line segment is a part of a straight line that has a length equal to the difference between the 2 points. The midpoint is the point halfway between the 2 points. Congruent line segments share a length.

Types of Angles

Angles are formed at the intersection of two lines. They can be less than 90 degrees, 90 degrees exactly, or greater than 90 degrees.

Acute angles are less than 90 degrees.
Right angles are 90 degrees.
Obtuse angles are greater than 90 degrees.
A straight angle is 180 degrees.

Two angles are called complementary if they sum to 90 degrees.
Supplementary angles are those that sum to 180 degrees.

Polygons

The combination of 3 or more lines to form a closed figure makes a polygon. The points at which the lines meet are called vertices, and the lines joining the vertices are called the sides. Triangles have 3 sides, quadrilaterals have 4 sides, and pentagons have 5 sides. A regular polygon contains angles and side lengths that are congruent.

To calculate the sum of the interior angles of a polygon, where n is the number of sides:

The perimeter of a polygon is the sum of its side lengths. The area of a polygon is the amount of space enclosed by the lines.

Triangles

Triangles are three-sided figures. The interior angles of a triangle sum to 180 degrees. Triangles can be acute, right, and obtuse. Right triangles contain a long side, called the hypotenuse, connecting two shorter sides .

The perimeter of a triangle is the distance around the figure.
The area is $\frac{1}{2} b \cdot h$ where b is the base of the triangle, and h is the height.

Right triangles exhibit special properties. The Pythagorean theorem enables the distance between two points (the hypotenuse of a right triangle) to be calculated, it is defined as:

$a^2 + b^2 = c^2$, where a and b are the shorter side lengths, and c is the hypotenuse.

There are 2 special right triangles to be familiar with- the 45, 45, 90 triangle and the 30, 60, 90 triangle.

The 45, 45, 90 right triangle has side lengths in a ratio of $x:x:x\sqrt{2}$.
The 30, 60, 90 right triangle has side lengths in a ratio of $x, x\sqrt{3}, 2x$.

Similar and congruent triangles can be solved by setting up proportions and cross multiplication.

Congruent versus Similar

Two figures are congruent if they are exactly the same shape and size.
Two figures are similar if they are the same shape but not the same size.
Both congruent and similar figures are proportional in size.

Quadrilaterals are polygons with four sides.

Squares have 4 equal sides and 4 right triangles. Rectangles have 2 pairs of equal sides and 4 right triangles. Parallelograms have parallel sides and congruent opposite angles and the sum of the angles along a line equals 180 degrees. A trapezoid contains 2 parallel lines.

The perimeter of a quadrilateral is the distance around the figure. The area is $bh$ for squares, rectangles, and parallelograms. The area is $\frac{1}{2} (b_1 + b_2) \cdot h$, where b is the base and h is the height.

A circle is a set of points equidistant from a center point.
The distance from the center point to a point along the circle is the radius.
The diameter is a line through the center of the circle that connects to points on the circle.
Lines that connect two points on the circle without passing through the center are chords.

The perimeter of a circle is called the circumference.
The formula for the circumference is: $C = 2 \cdot \pi \cdot r$ where r is the radius of the circle.
A portion of the circumference is called an arc.
The relationship between an arc, the radius, and the angle forming the arc is:

$s = \theta \cdot r$ where s is the arc length, $\theta$ is the angle (in radians), and r is the radius.

A circle contains 360 degrees, which is equal to 2$\pi$ radians.
To convert degrees to radians, multiply the degree measurement by $\frac{\pi}{180}$.
To convert radians to degrees, multiply by $\frac{180}{\pi}$.

The area of a circle is: $A = \pi \cdot r^2$, where r is the radius.

A line tangent to a circle intersects the circle exactly once and forms a 90 degree angle with the radius of the circle.

A figure inscribed inside of a circle has its vertices along the circumference.
Concentric circles share a center.

3-D Figures

When an extra dimension is added to a 2-D figure, a 3-D figure is formed. These figures possess a volume, which is the amount of space enclosed in the figure.

A rectangular prism has a length, width and height.
The volume of this figure is $V = lwh$.
The surface area of a rectangular prism is the sum of the areas of each face.

A circular cylinder is formed from 2 congruent circles with a surface along the circumferences that form parallel lines from one circle to the other.
The volume of this figure is $\pi \cdot r^2 \cdot h$.
The surface area is: $2\pi \cdot r^2 + 2\pi \cdot r \cdot h$.