Lesson 1 Go the Distance Develop Understanding

Learning Focus

Find the distance between two points in the coordinate plane.

Find the perimeter of a geometric figure in the coordinate plane.

How can I find the distance between two points if they’re not on vertical or horizontal lines?

Open Up the Math: Launch, Explore, Discuss

The performances of the Crawford High School drill team are very popular during half-time at the school’s football and basketball games. When the Crawford High School drill team choreographs the dance moves that they will do on the football field, they lay out their positions on a grid like the one shown:

In one of their dances, they plan to make patterns holding long, wide ribbons that will span from one dancer in the middle to six other dancers. On the grid, their pattern looks like this:

The question the dancers have is how long to make the ribbons. Gabriela ( $G$ ) is standing in the center, and some dancers think that the ribbon from Gabriela ( $G$ ) to Courtney ( $C$ ) will be shorter than the one from Gabriela ( $G$ ) to Brittney ( $B$ ).

1.

How long does each ribbon need to be?

2.

Explain how you found the length of each ribbon.

When they have finished with the ribbons in this position, they are considering using them to form a new pattern like this:

3.

Will the ribbons they used in the previous pattern be long enough to go between Brittney ( $B$ ) and Courtney ( $C$ ) in the new pattern? Explain your answer.

Gabriela notices that the calculations she is making for the length of the ribbons remind her of math class. She says to the group, “Hey, I wonder if there is a process that we could use like what we have been doing to find the distance between any two points on the grid.” She decides to think about it like this:

“I’m going to start with two points and draw the line between them that represents the distance that I’m looking for. Since these two points could be anywhere, I named them $A (x_{1}, y_{1})$ and $B (x_{2}, y_{2})$ . Hmmmmm. . . when I figured the length of the ribbons, what did I do next?”

4.

Think back on the process you used to find the length of the ribbon and write down your steps here, in terms of $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ .

5.

Use the process you came up with in problem 4 to find the distance between two points located far enough away from each other that using your formula from problem 4 is more efficient than graphing and counting. For example, find the distance between $(- 11, 25)$ and $(23, - 16)$ .

6.

Use your process to find the perimeter of the hexagon pattern shown in problem 3.

Ready for More?

Find as many points as you can that are $3$ units away from the point $(1, 1)$ . Plot each of the points along with $(1, 1)$ . What do you notice about the graph of the points?

Takeaways

Steps in words	Steps in symbols

The Distance Formula:

Finding the perimeter of a geometric figure on the coordinate plane:

Lesson Summary

In this lesson, we learned to find the distance between two points. We used the Pythagorean theorem to develop a formula that could be used whenever we need to find the length of a segment between two points. The formula can be applied to find the length of the sides of a geometric figure in the coordinate plane to calculate the perimeter.

Retrieval

1.

Point $A$ is graphed in the coordinate plane.

Rotate $A$ $90 °$ counterclockwise about the origin, and label the new coordinates. The new point is $A^{'}$ . Compare the coordinates of $A$ with the coordinates of $A^{'}$ .
Rotate point $A$ $90 °$ clockwise about the origin. Label the new coordinates. The new point is $A^{″}$ . Compare the coordinates of $A$ with the coordinates of $A^{″}$ .

2.

Fill in the missing coordinates. Then find $A C$ and $C E$ .