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 further investigate the point of intersection of the perpendicular bisectors, by observing and then proving that the point is equidistant from the vertices. Students have the opportunity to practice writing the equation of a circle and then using that equation to reason about the vertices of a triangle. Monitor for students who verify by checking distances and students who verify by checking points in the equation.

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

Students combine their work from this lesson and the previous lesson on triangle centers. They plot 3 centers for the same triangle (medians, altitudes, and perpendicular bisectors) and observe that the centers are collinear. When students are working on their proofs of this observation, monitor for those who write an equation for the line going through 2 of the points then substitute the third point into the equation, and for others who look at the slopes between the points.

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.