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In this lesson we’ll define coordinate geometry and learn about formulas that are commonly used in coordinate geometry. You can then take a brief quiz to see what you learned.

Definition of Coordinate Geometry

There are all sorts of ways that we can find the measurements of lines and angles. We can use rulers to measure lines and protractors to measure angles. In coordinate geometry we can use graphs and coordinates to find measurements and other useful information about geometric figures.

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Let’s start by reviewing the features of coordinate graphs. A coordinate graph is a rectangular grid with two number lines called axes. The x-axis is the horizontal number line and the y-axis is the vertical number line. The axes intersect at the origin which is the point (0,0).

Calculating Slope, Distance, and Midpoint

With coordinate geometry, various geometric figures can be graphed using the coordinates of the figure’s vertices (corners). We can use these coordinates along with mathematical formulas to calculate the length, slope, and midpoint of the sides.

slope


distance


The subscripts of 1 and 2 in the formulas are used to distinguish the coordinates of each point. For example, x sub 2 is the x-coordinate for the second point and tells us where the point is along the x-axis. y sub 2 is the y-coordinate for the second point and tells us where the point is along the y-axis.Check out this example:Find the slope, length, and midpoint of side AB in triangle ABC.


coordgeom2


To find the slope of side AB we’ll substitute the coordinates of points A and B into the slope formula. It is helpful to label the coordinates with x sub 1, x sub 2, etc. to remember what values to substitute in the formula.

The first value for each point is x, the second value is y. Then we add the subscripts of 1 and 2 for the first and second points. Once the coordinates are labeled, we can substitute them into the slope formula.

coordgeom3


slope calculation
slopecalc


The slope of side AB is -5/3.

The length of side AB is the same as the distance between points A and B, so we can use the distance formula.

distcalc


When using the distance formula it’s very important to follow the order of operations correctly. We find that the distance between points A and B, which is also the length of side AB, equals the square root of 34.You should note that if we were asked to find an angle measurement of the triangle, we could use the length of the sides along with trigonometric formulas.Lastly, we’ll use the midpoint formula to find the midpoint of line AB.

The midpoint is the point located at the exact center of the line. Keep in mind that it is a point, so the solution will be coordinates, not a single value.

midptcalc


The point (2.5, 2.

5) is the midpoint of line AB.

Lesson Summary

Coordinate geometry allows us to find measurements of geometric figures that can be graphed with coordinates. Also, remember that a coordinate graph is a rectangular grid with two number lines called axes: the x-axis is the horizontal number line and the y-axis is the vertical number line. The axes intercept at the origin, which is the point (0,0). Instead of relying on a ruler and protractor, we can use mathematical formulas to find the slopes, distances (or lengths), and midpoints of lines, which all have their own formulas.

Formulas for Coordinate Geometry

Coordinate geometry definition

Midpoint

midpoint

Distance

distance

Slope

slope

Learning Outcomes

When you are finished, you should be able to:

  • Describe a coordinate graph and name the two axes
  • Calculate the slope and midpoint of a line using the correct formulas
  • Determine the distance between two points using the appropriate formula

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