Finding Distances in the Coordinate Plane

Student Summary

We can use the Pythagorean Theorem to find the distance between any two points in the coordinate plane.

For example, if the coordinates of point AA are (-2,-3)(\text-2,\text-3), and the coordinates of point BB are (-8,4)(\text-8,4), the distance between them is also the length of line segment ABAB. It is a good idea to plot the points first.

The graph of a line segment in the coordinate plane with the origin labeled “O”. 
The graph of a line segment in the coordinate plane with the origin labeled “O”. On the x-axis, the numbers negative 8 through 0 are indicated. On the y-axis, the numbers negative 3 through 4 are indicated. The line segment begins to the left of the y axis and above the x axis at the point labeled B where point B has coordinates negative 8 comma 4. The line segment slants downward and to the right, crosses the x axis and ends at the point labeled A. Point A has coordinates negative 2 comma negative 3.

Think of the segment ABAB as the hypotenuse of a right triangle. The legs can be drawn in as horizontal and vertical line segments.

A triangle is graphed in the coordinate plane with the origin labeled “O”. 
A triangle is graphed in the coordinate plane with the origin labeled “O”. On the x-axis, the numbers negative 8 through negative one are indicated. On the y-axis, the numbers negative 3 through 4 are indicated. Two of the vertices, point A and point B, of the triangle are labeled. Point A is located at negative 2 comme negative 3 and point B is located at negative 8 comma 4. A vertical line is drawn from Point B directly down and a horizontal line is drawn from Point A to the left until the two lines meet creating the third vertex of the triangle. The two lines meet at the point with coordinates negative 8 comma negative 3. The vertical line is labeled with the text "the absolute value of four minus negative three equals 7". The horizontal line is labeled with the text "the absolute value of -8 minus -2 equals 6."

The length of the horizontal leg is 6, which can be seen in the diagram. This is also the distance between the xx-coordinates of AA and BB (∣-8−-2∣=6|\text-8-\text-2|=6).

The length of the vertical leg is 7, which can be seen in the diagram. This is also the distance between the yy-coordinates of AA and BB (∣4−-3∣=7|4 - \text-3|=7).

Once the lengths of the legs are known, we use the Pythagorean Theorem to find the length of the hypotenuse, ABAB, which we can represent with cc: 

62+72=c236+49=c285=c285=c\begin{aligned} 6^2+7^2&=c^2 \\ 36+49&=c^2 \\ 85&=c^2 \\ \sqrt{85}&=c \\ \end{aligned}

This length is a little longer than 9, since 85 is a little longer than 81. Using a calculator gives a more precise answer, 85≈9.22\sqrt{85} \approx 9.22.

Visual / Anchor Chart

Standards

Addressing
8.G.8

8.G.B.8

8.G.8

8.G.B.8

8.G.8

8.G.B.8

Building Toward
8.G.8

8.G.B.8