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GMAT Test: Online Prep and Review21 chapters | 173 lessons | 9 flashcard sets

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Lesson Transcript

Instructor:
*Melanie Olczak*

Melanie has taught high school Mathematics courses for the past ten years and has a master's degree in Mathematics Education.

This lesson will provide definitions and examples of coordinate geometry problems. Formulas and terms that are used in coordinate geometry will be explained.

Much of geometry can be done in the coordinate plane. It's often easier to find properties and show things when placing an object in the coordinate plane because we can use points to describe location, size, slope, and other properties.

The **coordinate plane** is a grid that has horizontal and vertical axes. The ** x-axis** is the horizontal axis and the

The *x* and *y* axes intersect and divide each other into positive and negative sections. We can graph a point (*x*,*y*) on a coordinate plane by first plotting the *x* coordinate, which is to the right if it's positive and to the left if it's negative. Then we plot the *y* coordinate, going up if it's positive and down if it's negative.

For example, let's plot the points C(-3, 6) and D(4, -2). First, we'll work with point C. The first number is the *x*-value, which is negative, so we will move to the left 3. The second number is the *y*-value so we will move up 6. Now, let's look at D. The first number is positive so we'll move 4 units to the right. The second number is negative, so we'll move 2 units down.

The coordinate plane is divided into four quadrants.

The first quadrant is where both *x* and *y* are positive. In the second quadrant, *x* is negative and *y* is positive. In the third quadrant, both *x* and *y* are negative. In the fourth quadrant, *x* is positive and *y* is negative.

We can graph points in the coordinate plane and find the distance between those two points by using the Pythagorean Theorem. The **Pythagorean Theorem** states that if we have a right triangle, the sum of the squares of the two legs is equal to the sum of the square of the hypotenuse, read in formula form as *a*^2 + *b*^2 = *c*^2, as you can see below:

The **hypotenuse** is the longest side of the right triangle, and it's the side that is across from the right angle.

If we have the points *F*(2, 1) and *E*(-3, 4), we can find the distance by using the Pythagorean Theorem. First, we'll graph the points in the coordinate plane.

Next, we will draw a right triangle, with the hypotenuse as the segment *EF* that we are trying to find.

Now, we can plug these points into the formula and solve for the hypotenuse, *c*.

Therefore, the length of segment *EF* is 5.82 units.

Using the points *A*(2, 1) and *B*(-3, 4), we can find the slope of the line *AB*. The **slope** of a line is the rate of change in the *y*-values over the change in the *x*-values. Typically, we refer to the slope as the rise over the run.

There are two ways to find the slope: we can use the graph, or we can use the points and find it algebraically.

If we're using the graph, remember that we read from left to right. As you can see, from left to right this line goes down.

Since the line goes down, the slope is negative. How far do we go down from *B* to *A*? Let's count on the graph. Since *B* is at 4 and *A* is at 1, we must go down 3. Next we find the run. How far do we run to the right? *B* is at -3 and *A* is at 2, so that's a total of 5 units. The slope is then -3/5.

Now let's use the slope formula which, as you can see, is:

To find the slope using the formula, we plug in the coordinates from points *A* and *B*, and then we simplify. Since *A* is the first point, we will use the coordinates of *A* as (*x*1, *y*1) and the coordinates of *B* as (*x*2, *y*2). However, if we were to switch the two points, we would still get the same slope. We get:

Notice we got the same slope that we did just by counting. When there's a graph provided, it is often easier to count. However, when there's no graph, it might be easier to plug numbers into the formula.

A **function** is a graph where every input *x* has only one output *y*. This means that each *x* value has only one corresponding *y* value. We can use what we call the **vertical line test**, or the pencil test, to see if a graph is a function. Take a pencil, line it up vertically, and move it from left to right across the graph. If the pencil crosses the graph at more than one point, the graph is not a function.

The pencil here will only cross the line one point at a time. Therefore, it's a function.

In this example, the pencil crosses the graph at two points, so it's not a function.

We can use the graphs of functions to find the *x*-intercepts and *y*-intercepts. An ** x-intercept** is the point where the graph of a function crosses the

Let's look at some examples:

In this example, we have a line, or a **linear function**. A line can only cross the *x*-axis in one place, which means it only has one solution.

*x*-intercept: -1

*y*-intercept: -1

In this example, we have a parabola, or a **quadratic function**. A parabola can have at most two *x*-intercepts.

*x*-intercepts: -1, 1

*y*-intercepts: -1

In this example, we have a cubic function. It has at most three *x*-intercepts.

*x*-intercepts: -3, -0.5, 2

*y*-intercept: -3

The **coordinate plane** is a grid where we can graph points based on location. The coordinate plane is separated into four quadrants. The quadrants start where both *x* and *y* are positive and are labeled in a counterclockwise direction.

We can use the **Pythagorean Theorem**, which states that if we have a right triangle the sum of the squares of the two legs is equal to the sum of the square of the hypotenuse, to find the length of the **hypotenuse** in a right triangle, which is the longest side of the right triangle. The sum of the squares of the legs is equal to the square of the hypotenuse. We can use the Pythagorean Theorem to find the distance between two points in the coordinate plane by drawing a right triangle with the missing side as the hypotenuse.

We can also find the slope of a line in the coordinate plane by using the **slope formula**:

We can graph functions in the coordinate plane and find their *x* and *y*-intercepts. **Functions** are graphs where each *x* value has only one *y* value. The ** x-intercept** is the point where a graph crosses the

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GMAT Test: Online Prep and Review21 chapters | 173 lessons | 9 flashcard sets

- What is a Line Segment in Geometry? - Definition, Formula & Example 3:53
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- Volumes of Shapes: Definition & Examples 4:21
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