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 pairs of lines on the solution or display some of the images shown here. Invite students to discuss any other observations. (A 90-degree rotation means that the original and image lines intersect at a 90-degree angle, which means that 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 who make different conjectures about the slopes of perpendicular lines. Here are some approaches students may take, ordered from use of less precise to more precise mathematical language:

• Describe the structure of the pairs of slopes using informal language such as “flipped” or “upside down.”
• Describe the structure of the pairs of slopes using terms like “inverse,” “opposite,” or “reciprocals.”
• State that the product of each pair of slopes is -1.

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 Activity Synthesis.