# Lesson 17

Lines in Triangles

### Lesson Narrative

The goal of this lesson is to hone the algebra skills students encountered during this unit. The lesson has a warm-up and an activity followed by 4 optional activities. There are several ways to assemble the optional activities into a lesson.

Students begin by investigating the intersection of a specific triangle’s altitudes. From this point there are a few paths available. Students could repeat this process for perpendicular bisectors and then observe the intersection point is the circumcenter to review the equation of a circle. They could skip the circle aspect and instead observe the intersection points of the medians, altitudes, and perpendicular bisectors are collinear, all falling along the Euler Line. Or students could skip perpendicular bisectors entirely and instead review writing equations of parallel and perpendicular lines by drawing a tessellation. Finally, this could be a multi-day lesson with all the activities included.

Triangle centers were indirectly explored in the Constructions and Rigid Transformations unit when students partitioned a region into areas closest to 3 given points. They will be revisited in the context of circumscribed and inscribed circles in a subsequent unit. That unit will approach triangle centers from a constructions point of view and will not rely on the information in these optional activities.

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.

### Learning Goals

Teacher Facing

• Use coordinates to see altitudes are concurrent.

### Student Facing

• Let’s investigate more special segments in triangles.

### Required Preparation

Toolkits are needed for tracing paper: 2 pieces per student if completing the perpendicular bisector activity, or 1 piece per student if not.

1 sheet of graph paper per student is needed if completing the tessellation activity.

### Student Facing

• I can determine the point where the altitudes of a triangle intersect.

Building On