Lesson 2 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
as the hypotenuse. You’ll need a vertical segment, , from point to point on the -axis and a horizontal segment from point to point . These are often called slope triangles. Based on the slope triangle that you have drawn, what is the slope of ? Now, rotate the slope triangle
counterclockwise about the origin. What are the coordinates of the image of point ? Using this new point,
, draw a slope triangle with hypotenuse . Based on the slope triangle, what is the slope of the line ?
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
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
and are constructed to be perpendicular. Start by labeling a point
on the line . Label the coordinates of
. Draw the slope triangle from point
. Label the lengths of the sides of the slope triangle using variables like
and for the run and the rise.
5.
What is the slope of line
Rotate point
6.
Draw the slope triangle from point
7.
What is the slope of line
8.
What is the relationship between the slopes of line
9.
Is the relationship between the slopes changed if the intersection between line
10.
Is the relationship between the slopes changed if lines
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
12.
Write a proof of the theorem: If two lines are perpendicular, the slopes are negative reciprocals.
Given:
Let
Prove:
Now think about parallel lines like the ones shown:
Points
13.
Draw the slope triangle from point
to point . What is the slope of ? Translate line
so that point coincides with point . Why will and lie on line ? Draw the slope triangle from point
to point . What are the coordinates of and ? What is the slope of
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
Takeaways
Given:
Prove:
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.
1.
Solve for
2.
A line passes through the points
standard form: