This lesson provides additional practice for students to find equations of lines in the coordinate plane, including parallel and perpendicular lines. The lesson has a Warm-up and an activity followed by two optional activities. 

Students begin by investigating the intersection of a specific triangle’s altitudes. They notice that the altitudes of a triangle coincide, and find the equations of each of the lines along which the altitudes lie. The first optional activity repeats the process for perpendicular bisectors, giving students the opportunity to compare perpendicular bisectors and altitudes of a triangle.

The second optional activity gives students an opportunity to create tessellations of right triangles, rectangles, and parallelograms, then to find equations that define their figures in an open-ended exploration of parallel and perpendicular lines in the plane. 

Triangle centers were indirectly explored in the “Constructions and Rigid Transformations” unit when students partitioned a region into areas closest to three given points. They will be revisited in the context of circumscribed and inscribed circles in a later course. That unit will approach triangle centers from a constructions point of view and will not rely on the information in this optional lesson.

In all cases, students are working on seeing structure (MP7) as they recognize the appropriate equations to use in each situation.

Technology isn’t required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.