Lesson 3 Slippery Slopes Solidify Understanding

Learning Focus

Prove slope relationships between parallel lines and perpendicular lines.

How do we know that the slopes of any two perpendicular lines are negative reciprocals? Is it always true or sometimes true?

Open Up the Math: Launch, Explore, Discuss

While working with right triangles previously, you examined several examples that led to the conclusion that the slopes of perpendicular lines are negative, or opposite, reciprocals. Your goal now is to formalize this work into a proof. Let’s start by thinking about two perpendicular lines that intersect at the origin, like these:

1.

Start by drawing a right triangle with the segment $\overset{―}{O A}$ as the hypotenuse. You’ll need a vertical segment, $\overset{―}{A B}$ , from point $A$ to point $B$ on the $x$ -axis and a horizontal segment from point $B$ to point $O$ . These are often called slope triangles. Based on the slope triangle that you have drawn, what is the slope of $\overset{\leftrightarrow}{O A}$ ?
Now, rotate the slope triangle $90 °$ counterclockwise about the origin. What are the coordinates of the image of point $A$ ?
Using this new point, $A^{'}$ , draw a slope triangle with hypotenuse $\overset{―}{O A^{'}}$ . Based on the slope triangle, what is the slope of the line $\overset{\leftrightarrow}{O A^{'}}$ ?

2.

What is the relationship between these two slopes? How do you know?

3.

Is the relationship changed if the two lines are translated so that the intersection is at $(- 5, 7)$ ?

How do you know?

To prove a theorem, we need to demonstrate that the property holds for any pair of perpendicular lines, not just a few specific examples. It is often done by drawing a very similar picture to the examples we have tried, but using variables instead of numbers. Using variables represents the idea that it doesn’t matter which numbers we use, the relationship stays the same. Let’s try that strategy with the theorem about perpendicular lines having slopes that are negative reciprocals.

4.

Lines $ℓ$ and $m$ are constructed to be perpendicular.
Start by labeling a point $P$ on the line $ℓ$ .
Label the coordinates of $P$ .
Draw the slope triangle from point $P$ .
Label the lengths of the sides of the slope triangle using variables like $a$ and $b$ for the run and the rise.

5.

What is the slope of line $ℓ$ ?

Rotate point $P$ $90 °$ about the origin, label it $P^{'}$ , and mark it on line $m$ . What are the coordinates of $P^{'}$ ?

6.

Draw the slope triangle from point $P^{'}$ . What are the lengths of the sides of the slope triangle? How do you know?

7.

What is the slope of line $m$ ?

8.

What is the relationship between the slopes of line $ℓ$ and line $m$ ? How do you know?

9.

Is the relationship between the slopes changed if the intersection between line $ℓ$ and line $m$ is translated to another location? How do you know?

10.

Is the relationship between the slopes changed if lines $ℓ$ and $m$ are rotated?

11.

How do these steps demonstrate that the slopes of perpendicular lines are negative reciprocals for any pair of perpendicular lines?

In problems 6–9, you proved that the product of the slopes of two perpendicular lines intersecting at the origin is $- 1$ . You have also observed that if the intersection of the two perpendicular lines is not $(0, 0)$ , you can translate the intersection to the origin. By definition, all the angles and distances are preserved under such a translation. From there, the relationship that the product of the slopes is $- 1$ , as described above, would still be the same.

12.

Write a proof of the theorem: If two lines are perpendicular, the slopes are negative reciprocals.

Given: $ℓ ⊥ m$

Let $q_{1}$ be the slope of $ℓ$ and $q_{2}$ be the slope of $m$ .

Prove: $q_{1} = - \frac{1}{q_{2}}$

Now think about parallel lines like the ones shown: $m ∥ n$ .

Points $O$ , $P$ , and $Q$ all lie on line $n$ . Point $B$ is the $y$ -intercept of line $m$ .

13.

Draw the slope triangle from point $P$ to point $Q$ . What is the slope of $\overset{\leftrightarrow}{P Q}$ ?
Translate line $n$ so that point $O$ coincides with point $B$ . Why will $P^{'}$ and $Q^{'}$ lie on line $m$ ?
Draw the slope triangle from point $P^{'}$ to point $Q^{'}$ . What are the coordinates of $P^{'}$ and $Q^{'}$ ?
What is the slope of $\overset{\leftrightarrow}{P^{'} Q^{'}} ?$
Show how you know that these two parallel lines have the same slope, and explain why this proves that all parallel lines have the same slope.

Ready for More?

Find the equation of the perpendicular bisector of the segment with endpoints $(- 5, 6)$ and $(4, - 3)$ .

Takeaways

Given: $m ∥ n$ with point $O$ with coordinates $(0, 0)$ , point $P$ with coordinates $(a, b)$ , and point $Q$ with coordinates $(c, d)$ on $n$ . Let $q_{1}$ be the slope of $n$ and $q_{2}$ be the slope of $m$ .

Prove: $q_{1} = q_{2}$

Lesson Summary

In this lesson, we used transformations to prove that the slopes of perpendicular lines are negative reciprocals and the slopes of parallel lines are equal. To prove the theorems, we needed to write the lines and points so that they were general enough to cover all cases. When we used a specific point like the origin, we needed to make an argument that the relationship could be extended to any pair of lines that are parallel or perpendicular.

Retrieval

1.

Solve for $d$ .

$d^{2} - {(y_{2} - y_{1})}^{2} = {(x_{2} - x_{1})}^{2}$

2.

A line passes through the points $P (- 3, - 14)$ and $Q (17, - 19)$ . Use the given information to write the equation of the line in standard form.

standard form: $(A x + B y = C)$