# 5.1: Points & Lines & Slopes

**At Grade**Created by: CK-12

*This activity is intended to supplement Algebra I, Chapter 4, Lesson 3.*

## Problem 1 – Coordinates of Points

Open the *CabriJr* app by pressing **APPS**. Open a new file and make sure the axes are displayed. Place a point, \begin{align*}R\end{align*}, on the \begin{align*}x-\end{align*}axis and a point, \begin{align*}S\end{align*}, on the \begin{align*}y-\end{align*}axis. Display the coordinates of the points.

1. Explain what is common to all points on the \begin{align*}x-\end{align*}axis.

2. Explain what is common to all points on the \begin{align*}y-\end{align*}axis.

Delete points \begin{align*}R\end{align*} and \begin{align*}S\end{align*}. Place two points, \begin{align*}P\end{align*} and \begin{align*}Q\end{align*}, in the top right section. Drag the points around into different sections.

Complete the sentences by writing *positive* or *negative*.

3. A point is in Quadrant 1 (top right) when its \begin{align*}x-\end{align*}coordinate is ___________ and its \begin{align*}y-\end{align*}coordinate is ___________.

4. A point is in Quadrant 2 (top left) when its \begin{align*}x-\end{align*}coordinate is ___________ and its \begin{align*}y-\end{align*}coordinate is ___________.

5. A point is in Quadrant 3 (bottom left) when its \begin{align*}x-\end{align*}coordinate is ___________ and its \begin{align*}y-\end{align*}coordinate is ___________.

6. A point is in Quadrant 4 (bottom right) when its \begin{align*}x-\end{align*}coordinate is ___________ and its \begin{align*}y-\end{align*}coordinate is ___________.

Draw two lines through point \begin{align*}P\end{align*}, one perpendicular to the \begin{align*}x-\end{align*}axis and the other perpendicular to the \begin{align*}y-\end{align*}axis. Construct segments from point \begin{align*}P\end{align*} to the axes, and then hide the lines. Measure the length of each segment. Drag point \begin{align*}P\end{align*} and explore.

7. What is this relationship between the coordinates of point \begin{align*}P\end{align*} and the distances to each axis?

## Problem 2 – Lines, Equations, and Slopes

Delete or hide the segments and measurements. Draw a line connecting \begin{align*}P\end{align*} and \begin{align*}Q\end{align*}. Using tools from the Appearance menu find the equation and slope of the line.

Look for relationships between the slope and equation as you change the line by grabbing and dragging point \begin{align*}P\end{align*}, and then by grabbing and dragging the line itself.

8. When dragging the line by point \begin{align*}P\end{align*}, what is the relationship of the slope and the equation?

9. When dragging the line itself, what is changing in the equation?

10. Drag point \begin{align*}Q\end{align*} to the \begin{align*}y-\end{align*}axis. What is the relationship between point \begin{align*}Q\end{align*} and the equation of the line?

## Problem 3 – Slopes of Parallel and Perpendicular Lines

Open the Cabri Jr. file **PARALLEL**. Drag the lines by points \begin{align*}P\end{align*} and \begin{align*}Q\end{align*} and examine the slopes.

11. What can you say about the slopes of two parallel lines?

Open the Cabri Jr. file **PERPENDI**. Again drag the lines to investigate the relationship between the slopes.

12. What can you say about the slopes of two perpendicular lines?

Use the **Calculate** tool to see what happens when the slopes of two perpendicular lines are multiplied together. Select one slope measurement, press \begin{align*}\times\end{align*}, and select the other slope measurement. Move the product near the original expression.

Now, change the lines by grabbing and dragging point \begin{align*}P\end{align*}.

13. What do you observe about the product of the slopes?

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