Introduction to Coordinate Geometry Word Problems
Inspiration is needed in geometry, just as much as in poetry.
—ALEKSANDR PUSHKIN (1799–1837)
This lesson will cover the coordinate plane and important topics related to the coordinate plane, such as quadrants, slope, distance, and midpoint.
The Coordinate Plane
The coordinate plane is made up of two perpendicular number lines that intersect, or cross, at a point known as the origin. Every point in the coordinate plane has a location determined by its coordinates, (x,y), and the origin has coordinates (0,0). When these lines intersect, they create four sections called quadrants. The quadrants are numbered, usually with Roman numerals, starting with the upper righthand quadrant.
Tip:
The coordinates of points determine in which quadrant they are located. The first value in the parentheses is the xcoordinate and determines the number of units to move right (if a positive number) or left (if a negative number). The second value in the parentheses is the ycoordinate and determines the number of units to move up (if a positive number) or down (if a negative number).

An example of the coordinate plane with the origin and quadrants labeled is shown below.
Each point in the plane has a location given by two values in parentheses. To graph a point in the plane, start at the origin. Look at the first value. Move to the right that number of spaces if the first number is positive or to the left if the first number is negative. From there, look at the second value in parentheses. If the number is positive, move up that many spaces or down that many spaces if the number is negative.
For example, to graph the point A(4,5), start at the origin and move four units to the right and five units up. To graph the point B(2,–3), start at the origin and move two spaces to the right and three units down. To graph the point C(–3,1), move three units to the left and one unit up from the origin. To graph the point D(–1,–2), count one unit to the left and two units down from the origin. Each of these points is graphed in the figure.
Tip:
The formulas in coordinate geometry often use the values (x_{1},y_{1}) and (x_{2},y_{2}). Use the subscripts with x and y to keep track of each point.

Slope Formula
The slope of a line can be described as the steepness of the line. Lines that tilt up to the right have positive slope. Lines that tilt up to the left have negative slope. Horizontal lines have zero slope, and vertical lines have undefined slope. To find the slope of a line, first take any two points on the line. Then, to find the slope between the two points, use the formula
Substitute the coordinates of the points into the formula and evaluate.
Example
What is the slope between the points (4,2) and (–3,7)?
Read and understand the question. This question is looking for the slope between two points.
Make a plan. Use the formula
and substitute the coordinates of the two points. Use (4,2) as (x_{1},y_{1}) and (–3,7) as (x_{2},y_{2}).
Carry out the plan. The formula becomes
The slope is .
Check your answer. To check this result, substitute the values in the opposite order to make sure that the slope is the same. The formula is
This result is checking.
Distance Formula
Another common formula used in coordinate geometry is the distance formula. To find the distance between any two points, use the formula
Example
What is the distance between the two points (1,6) and (–2,2)?
Read and understand the question. This question is looking for the distance between two points.
Make a plan. Use the formula and substitute the coordinates of the two points. Use (1,6) as (x_{1},y_{1}) and (–2,2) as (x_{2},y_{2}).
Carry out the plan. The formula becomes
Evaluate the exponents and add the squares together.
Take the square root of 25 to get a distance of 5 units.
Check your answer. To check this result, substitute the values in the opposite order to make sure that the distance is the same. The formula is
units
This result is checking.
Midpoint Formula
The midpoint of two points is the location halfway between the points. To find the midpoint between any two points, use the formula .
Example
What is the midpoint between the two points (9,3) and (–5,1)?
Read and understand the question. This question is looking for the midpoint between two points.
Make a plan. Use the formula and substitute the coordinates of the two points. Use (9,3) as (x_{1},y_{1}) and (–5,1) as (x_{2},y_{2}).
Carry out the plan. The formula becomes. Add to get Simplify each fraction to get the midpoint of (2,2).
Check your answer. To check this result, substitute the values in the opposite order to make sure that the midpoint is the same. The formula is
This result is checking.
Find practice problems and solutions for these concepts at Coordinate Geometry Word Problems Practice Questions.
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From Math Word Problems in 15 Minutes A Day. Copyright © 2009 by LearningExpress, LLC. All Rights Reserved.