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# 6.1: Perpendicular Slopes

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Algebra I, Chapter 5, Lesson 4.

In this activity, you will explore:

• an algebraic relationship between the slopes of perpendicular lines
• a geometric proof relating these slopes

## Problem 1 – An Initial Investigation

Open the Cabir Jr. app by pressing APPS and choosing it from the menu. Press ENTER. Press any key to begin.

The calculator displays the Cabri Jr. window. Open the F1: File menu by pressing Y=\begin{align*}Y=\end{align*}. Arrow down to the Open… selection and press ENTER.

Choose figure PERP1 and press ENTER.

Two lines are displayed:

line L1\begin{align*}L1\end{align*} with a slope of m1\begin{align*}m1\end{align*} and line L2\begin{align*}L2\end{align*} with a slope of m2\begin{align*}m2\end{align*}.

Notice that the angle formed by the intersection of the lines measures 90\begin{align*}90^\circ\end{align*}; that is, the two lines are perpendicular.

Grab line L1\begin{align*}L1\end{align*} by moving the cursor over the point pressing ALPHA cursor turns into a hand to show that you have grabbed the point.

Rotate L1\begin{align*}L1\end{align*} by dragging the point using the arrow keys. Observe that as the slopes of the lines change, the two lines remain perpendicular. Explore the relationship between the slopes by answering the questions below.

1. Can you rotate L1\begin{align*}L1\end{align*} in such a way that m1\begin{align*}m1\end{align*} and m2\begin{align*}m2\end{align*} are both positive? Both negative?
2. Can you rotate L1\begin{align*}L1\end{align*} so that m1\begin{align*}m1\end{align*} or m2\begin{align*}m2\end{align*} equals ? If so, what is the other slope?
3. Can you rotate L1\begin{align*}L1\end{align*} so that m1\begin{align*}m1\end{align*} or m2\begin{align*}m2\end{align*} equals 1\begin{align*}1\end{align*}? If so, what is the other slope?
4. Rotate L1\begin{align*}L1\end{align*} so that m1\begin{align*}m1\end{align*} is a negative number close to zero. What can be said m2\begin{align*}m2\end{align*}?
5. Rotate L1\begin{align*}L1\end{align*} so that m1\begin{align*}m1\end{align*} is a positive number close to zero. What can be said about m2\begin{align*}m2\end{align*}?

## Problem 2 – A Closer Examination

Now that you have observed some of the general relationships between the slopes of two perpendicular lines, it is time to make a closer examination. Press 2nd\begin{align*}2^{nd}\end{align*} [MODE] to exit Cabri Jr.

Press PRGM to open the program menu. Choose PERP2 from the list and press ENTER twice to execute it.

Enter a slope of 2\begin{align*}2\end{align*} and press ENTER.

The program graphs a line L1\begin{align*}L1\end{align*} with the slope you entered and a line L2\begin{align*}L2\end{align*} that is perpendicular to L1\begin{align*}L1\end{align*}. m1\begin{align*}m1\end{align*} is the slope of L1\begin{align*}L1\end{align*} and m2\begin{align*}m2\end{align*} is the slope of L2\begin{align*}L2\end{align*}.

Press ENTER and the calculator prompts you for another slope. Use the graph to complete the following.

1. Enter to make the slope of L1\begin{align*}L1\end{align*} equal to . What is the slope of L2\begin{align*}L2\end{align*}?

2. What is the slope of L2\begin{align*}L2\end{align*} when the slope of L1\begin{align*}L1\end{align*} is 1\begin{align*}1\end{align*}?

3. What is the slope of L2\begin{align*}L2\end{align*} when the slope of L1\begin{align*}L1\end{align*} is 1\begin{align*}-1\end{align*}?

Enter other values for the slope of L1\begin{align*}L1\end{align*} and examine the corresponding slope of L2\begin{align*}L2\end{align*}. For each slope that you enter, m1\begin{align*}m1\end{align*} and its corresponding value of m2\begin{align*}m2\end{align*} are recorded in the lists L1\begin{align*}L_1\end{align*} and L2\begin{align*}L_2\end{align*}. To see a history of your “captured” values, enter a slope of 86\begin{align*}86\end{align*} to exit the program. Then press STAT and ENTER to enter the List Editor. The values of m1\begin{align*}m1\end{align*} are recorded in L1\begin{align*}L_1\end{align*} and the values of m2\begin{align*}m2\end{align*} are recorded in L2\begin{align*}L_2\end{align*}.

4. Conjecture a formula that relates the slope of two perpendicular lines. Enter your formula in the top of L3\begin{align*}L_3\end{align*} (with variable L1\begin{align*}L_1\end{align*}) to test your conjecture.

## Problem 3 – A Geometric Look

Start the Cabri Jr. app and open the file PERP3.

This figure shows another way to examine the slopes of perpendicular lines, geometrically. There should be two lines, L1\begin{align*}L1\end{align*} and L2\begin{align*}L2\end{align*}, with a slope triangle attached to each of them.

Grab line L1\begin{align*}L1\end{align*}, rotate it, and compare the rise/run triangles.

1. What do you notice about the two triangles?

## Problem 4 – The Analytic Proof

We now will analytically verify that two lines with slopes m1\begin{align*}m1\end{align*} and m2\begin{align*}m2\end{align*} are perpendicular if and only if m1m2=1\begin{align*}m1 \cdot m2 = -1\end{align*}.

(All of the following assumes m10\begin{align*}m1 \neq 0\end{align*}. What can be said about the case when m1=0\begin{align*}m1 = 0\end{align*}?)

Open the CabriJr file PERP4. This graph shows two perpendicular lines L1\begin{align*}L1\end{align*} and L2\begin{align*}L2\end{align*} with slopes m1\begin{align*}m1\end{align*} and m2\begin{align*}m2\end{align*} respectively, translated such that their point of intersection is at the origin. Refer to the diagram to answer the questions below.

1. What are the equations of these translated lines as shown in the diagram?
2. Let P\begin{align*}P\end{align*} be the point of intersection of line L1\begin{align*}L1\end{align*} and the vertical line x=1\begin{align*}x = 1\end{align*} and let Q\begin{align*}Q\end{align*} be the point of intersection of line L2\begin{align*}L2\end{align*} and the line x=1\begin{align*}x = 1\end{align*}. What are the coordinates of points P\begin{align*}P\end{align*} and Q\begin{align*}Q\end{align*}?
3. Use the distance formula to compute the lengths of OP¯¯¯¯¯¯¯¯,OQ¯¯¯¯¯¯¯¯\begin{align*}\overline{OP},\overline{OQ}\end{align*}, and PQ¯¯¯¯¯¯¯¯\begin{align*}\overline{PQ}\end{align*}.(Your answers should again be in terms of m1\begin{align*}m1\end{align*} and m2\begin{align*}m2\end{align*}.)
4. Apply the Pythagorean Theorem to triangle POQ\begin{align*}POQ\end{align*} and simplify. Does this match your conjecture from Problem 2?

## Problem 5 – Extension Activity #1

The CabriJr file PERP5 shows a circle with center O\begin{align*}O\end{align*} and radius OR\begin{align*}OR\end{align*}. Line T\begin{align*}T\end{align*} is tangent to the circle at point R\begin{align*}R\end{align*}.

The slopes of line T\begin{align*}T\end{align*} and segment OR\begin{align*}OR\end{align*} are shown (mT\begin{align*}mT\end{align*} and mOR\begin{align*}mOR\end{align*}, respectively.)

Your first task is to calculate 1mOR\begin{align*}\frac{1}{mOR}\end{align*}. Activate the Calculate tool, found in the F5: Appearance menu.

Move the cursor over 1\begin{align*}1\end{align*} and press ENTER. Repeat to select mOR\begin{align*}mOR\end{align*}, the slope of the segment OR\begin{align*}OR\end{align*}.

Press / to divide the two numbers. Drag the quotient to a place on the screen where you can see it clearly and press ENTER again to place it.

1. Grab point R\begin{align*}R\end{align*} and drag it around the circle. Observe the changing values of mT,mOR\begin{align*}mT, mOR\end{align*}, and 1mOR\begin{align*}\frac{1}{mOR}\end{align*}. What can you conjecture about the relationship between a tangent line to a circle and its corresponding radius?

## Problem 6 – Extension activity #2

The CabriJr file PERP6 shows a circle with an inscribed triangle QPR\begin{align*}QPR\end{align*}. The segment QR\begin{align*}QR\end{align*} is a diameter of the circle.

The slopes of segments PR\begin{align*}PR\end{align*} and PQ\begin{align*}PQ\end{align*} are shown (mPR\begin{align*}mPR\end{align*} and mPQ\begin{align*}mPQ\end{align*}, respectively.)

Compute 1mPQ\begin{align*}\frac{1}{mPQ}\end{align*} using the Calculate tool.

1. Grab point P\begin{align*}P\end{align*}, drag it around the circle, and examine the changing values. What can you conjecture about a triangle inscribed in a circle such that one side is a diameter?

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