Students construct the altitudes of a triangle and observe that all three segments intersect at a single point.

If necessary, remind students that the definition of altitude is a line through a vertex perpendicular to the opposite side of the triangle.

If students aren’t sure how to fold an altitude, ask them how they would fold the perpendicular bisector of a side. Then, how can they modify that fold to create an altitude?

If students struggle to visualize what the altitudes will look like and therefore have trouble folding them, an index card can be a useful way to help “see” the 90-degree angle.

Ask students to look at the triangles of students near them. What do they notice? (Everyone’s altitudes seem to intersect at one point.)

Tell students they’ll verify if this is true for all triangles in the next activity.

In this activity students use the structure of the coordinate plane to examine the observation that the altitudes of a triangle all intersect at a single point. Students practice writing equations for perpendicular lines as they represent altitudes algebraically. Then they solve the system of equations (a fairly simple system with as one equation) to show that the altitudes all intersect at a single point.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Students repeat the process from the Warm-up and previous activity, this time studying perpendicular bisectors. They continue studying the same triangle, so both the structure and some details (slopes) carry through to this activity.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Students saw tessellations earlier in this course when they made Voronoi diagrams and then colored in the new tessellation created by the perpendicular bisectors. In this activity they will draw their own tessellation and practice writing equations of parallel, perpendicular, and intersecting lines.