Lesson 11

Students use a coordinate transformation rule to perform a 90-degree rotation. In upcoming activities, students will use transformations to prove the slope criterion for perpendicular lines.

Invite students to share the effects they came up with. Listen for terms and descriptions such as rotation, 90 degrees, counterclockwise, the origin is the center, and perpendicular lines. If perpendicularity is not mentioned by students, draw the lines \(AD\) and \(AD’\) on the solution or display the image shown here. Invite students to discuss any other observations. (A 90-degree rotation means the original and image lines intersect at a 90-degree angle, which means the lines are perpendicular.)

Students use the results of several slope calculations to make a conjecture that the slopes of perpendicular lines are opposite reciprocals. In an upcoming activity, they will prove their conjecture.

Making a spreadsheet available gives students an opportunity to choose appropriate tools strategically (MP5).

Monitor for students whose conjecture is that the product of the slopes is -1, and for those who mention something about the structure of the two slopes, such as the fact that they have opposite signs or that they are “flipped.”

Students may need to be reminded that a product is the result of multiplication.

Select previously identified students to share in this order:

• First, invite a student whose conjecture involved the structures of the slopes to share their idea. Challenge students to use precise language. For example, if they say the slopes are “flipped” or “upside down,” ask them if there is a mathematical term for this concept (reciprocal). Students may want to describe the slopes of perpendicular lines as negative reciprocals. Opposite reciprocals is clearer language—if the original slope is negative (such as \(\text-\frac52\)), we can avoid the awkward \(\text-\left(\text-\frac{5}{2}\right)\) and instead jump directly to \(\frac52\).
• Next, invite a student whose conjecture involved the product of the slopes being -1 to share. Ask students if this conjecture is different from the previous one. Explore this idea by presenting a slope of \(\frac{a}{b}\) to students, and asking them to provide the opposite reciprocal (\(\text-\frac{b}{a}\)). Instruct students to find the product of these fractions (-1). The conjectures are saying the same thing, because the value that multiplies with \(\frac{a}{b}\) to create -1 is \(\text-\frac{b}{a}\).

If it hasn’t come up, tell students there is at least one exception that doesn’t fit the conjecture: a pair of horizontal and vertical lines. Because a vertical line has no slope, the idea of an opposite reciprocal doesn’t make sense.

Finally, challenge students to calculate the opposite reciprocal of several values, such as \(\frac47\), -5, and \(\frac13\).

Students use transformation arguments to prove that the slopes of perpendicular lines that pass through the origin are opposite reciprocals. The proof is extended to all pairs of non-vertical and non-horizontal perpendicular lines in the whole-class synthesis.

If students struggle to find the first slope, suggest that they draw a picture of a line passing through the origin and choose a point on the line to label \((a,b)\).

Ask students if their proof for lines through the origin can be extended to other lines that do not pass through the origin. (Yes. For a line that doesn’t pass through the origin, we could first translate it so it passes through the origin. Translation of lines results in parallel lines, so this translation would not affect the slope. The proof doesn’t apply if we start with a vertical line, though, because it has no slope.)

Then, ask students what the difference is between what they did in the previous activity and what they did in this one. (In the previous activity, we showed that the slopes of perpendicular lines were opposite reciprocals for a few specific cases. In this activity, we showed that this is true for all pairs of non-vertical and non-horizontal lines.)

Tell students that the converse is also true: If 2 lines have slopes that are opposite reciprocals, then they are perpendicular. Ask students to add this theorem to their reference charts as you add it to the class reference chart:

Lines are perpendicular if and only if their slopes are opposite reciprocals. (Theorem)

Finally, invite students to write the equation of a line that passes through the point \((\text-9, 3)\) and is perpendicular to the line given by \(y-5=\frac27 (x+2)\). The result is \(y-3=\text-\frac72 (x-(\text-9))\).