4.3: Perpendicular Slopes
This activity is intended to supplement Geometry, Chapter 3, 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 \begin{align*}F1\end{align*}: File menu by pressing \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 \begin{align*}L1\end{align*} with a slope of \begin{align*}m1\end{align*} and line \begin{align*}L2\end{align*} with a slope of \begin{align*}m2\end{align*}.
Notice that the angle formed by the intersection of the lines measures \begin{align*}90^\circ\end{align*}; that is, the two lines are perpendicular.
Grab line \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 \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.
- Can you rotate \begin{align*}L1\end{align*} in such a way that \begin{align*}m1\end{align*} and \begin{align*}m2\end{align*} are both positive? Both negative?
- Can you rotate \begin{align*}L1\end{align*} so that \begin{align*}m1\end{align*} or \begin{align*}m2\end{align*} equals 0? If so, what is the other slope?
- Can you rotate \begin{align*}L1\end{align*} so that \begin{align*}m1\end{align*} or \begin{align*}m2\end{align*} equals 1? If so, what is the other slope?
- Rotate \begin{align*}L1\end{align*} so that \begin{align*}m1\end{align*} is a negative number close to zero. What can be said \begin{align*}m2\end{align*}?
- Rotate \begin{align*}L1\end{align*} so that \begin{align*}m1\end{align*} is a positive number close to zero. What can be said about \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 \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 and press ENTER.
The program graphs a line \begin{align*}L1\end{align*} with the slope you entered and a line \begin{align*}L2\end{align*} that is perpendicular to \begin{align*}L1\end{align*}. \begin{align*}m1\end{align*} is the slope of \begin{align*}L1\end{align*} and \begin{align*}m2\end{align*} is the slope of \begin{align*}L2\end{align*}.
Press ENTER and the calculator prompts you for another slope. Use the graph to complete the following.
1. Enter 0 to make the slope of \begin{align*}L1\end{align*} equal to 0. What is the slope of \begin{align*}L2\end{align*}?
2. What is the slope of \begin{align*}L2\end{align*} when the slope of \begin{align*}L1\end{align*} is 1?
3. What is the slope of \begin{align*}L2\end{align*} when the slope of \begin{align*}L1\end{align*} is -1?
Enter other values for the slope of \begin{align*}L1\end{align*} and examine the corresponding slope of \begin{align*}L2\end{align*}. For each slope that you enter, \begin{align*}m1\end{align*} and its corresponding value of \begin{align*}m2\end{align*} are recorded in the lists \begin{align*}L_1\end{align*} and \begin{align*}L_2\end{align*}. To see a history of your “captured” values, enter a slope of 86 to exit the program. Then press STAT and ENTER to enter the List Editor. The values of \begin{align*}m1\end{align*} are recorded in \begin{align*}L_1\end{align*} and the values of \begin{align*}m2\end{align*} are recorded in \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 \begin{align*}L_3\end{align*} (with variable \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, \begin{align*}L1\end{align*} and \begin{align*}L2\end{align*}, with a slope triangle attached to each of them.
Grab line \begin{align*}L1\end{align*}, rotate it, and compare the rise/run triangles.
- What do you notice about the two triangles?
Problem 4 – The analytic proof
We now will analytically verify that two lines with slopes \begin{align*}m1\end{align*} and \begin{align*}m2\end{align*} are perpendicular if and only if \begin{align*}m1 \cdot m2 = -1\end{align*}.
(All of the following assumes \begin{align*}m1 \ne 0\end{align*}. What can be said about the case when \begin{align*}m1 = 0\end{align*}?)
Open the CabriJr file PERP4. This graph shows two perpendicular lines \begin{align*}L1\end{align*} and \begin{align*}L2\end{align*} with slopes \begin{align*}m1\end{align*} and \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.
- What are the equations of these translated lines as shown in the diagram?
- Let \begin{align*}P\end{align*} be the point of intersection of line \begin{align*}L1\end{align*} and the vertical line \begin{align*}x = 1\end{align*} and let \begin{align*}Q\end{align*} be the point of intersection of line \begin{align*}L2\end{align*} and the line \begin{align*}x = 1\end{align*}. What are the coordinates of points \begin{align*}P\end{align*} and \begin{align*}Q\end{align*}?
- Use the distance formula to compute the lengths of \begin{align*}\overline{OP}\end{align*}, \begin{align*}\overline{OQ}\end{align*}, and \begin{align*}\overline{PQ}\end{align*}. (Your answers should again be in terms of \begin{align*}m1\end{align*} and \begin{align*}m2\end{align*}.)
- Apply the Pythagorean Theorem to triangle \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 \begin{align*}O\end{align*} and radius \begin{align*}OR\end{align*}. Line \begin{align*}T\end{align*} is tangent to the circle at point \begin{align*}R\end{align*}.
The slopes of line \begin{align*}T\end{align*} and segment \begin{align*}OR\end{align*} are shown (\begin{align*}mT\end{align*} and \begin{align*}mOR\end{align*}, respectively.)
Your first task is to calculate \begin{align*}\frac{1}{mOR}\end{align*}. Activate the Calculate tool, found in the \begin{align*}F5\end{align*}: Appearance menu. Move the cursor over 1 and press ENTER. Repeat to select \begin{align*}mOR\end{align*}, the slope of the segment \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.
- Grab point \begin{align*}R\end{align*} and drag it around the circle. Observe the changing values of \begin{align*}mT\end{align*}, \begin{align*}mOR\end{align*}, and \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 \begin{align*}QPR\end{align*}. The segment \begin{align*}QR\end{align*} is a diameter of the circle.
The slopes of segments \begin{align*}PR\end{align*} and \begin{align*}PQ\end{align*} are shown (\begin{align*}mPR\end{align*} and \begin{align*}mPQ\end{align*}, respectively.)
Compute \begin{align*}\frac{1}{mPQ}\end{align*} using the Calculate tool.
- Grab point \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?