Lesson 17

Students construct the altitudes of a triangle and observe that all 3 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 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 \(x=2\) 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).

If students aren’t sure how to find the slopes of the altitudes, ask them about the relationship between the slope of a side and the slope of the altitude through that side. (The product of the slopes is -1 since they are perpendicular.)

If students struggle to verify algebraically that \(H\) is the point of intersection, ask them what’s true if a particular point is on 3 different lines (the lines must intersect at that point, unless they coincide). How can students test if \(H\) is on each line?

As in the previous activity, an index card can be a useful too to help visualize the altitudes.

Ask students what the relationship is between the slope of a side and the slope of the altitude through that side. (The product of the slopes is -1 since they are perpendicular.) Invite students to share strategies for verifying their coordinates of \(H\). (Sample response: Start with \(x=2\), find the \(y\)-value using one equation, and test that point in the other equation.)

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).

Draw another triangle on tracing paper. Fold the perpendicular bisector of each side. What do you notice? (These segments also intersect at a single point.) Remind students that the definition of a perpendicular bisector is a line passing through the midpoint of a segment, perpendicular to that segment.

Use the same slopes from the previous activity.

If students confuse altitudes, medians, and perpendicular bisectors, remind them that altitudes and medians must go through the triangle’s vertices, but the perpendicular bisectors don’t necessarily do so.

Invite students to share strategies for verifying their coordinates of \(P\). (Sample response: Start with \(x=4\), find the \(y\)-value using one equation, and test that point in the other equation.)

Students further investigate the point of intersection of the perpendicular bisectors by observing and then proving 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).

Invite the previously selected students to share their methods of verification. “What is the definition of a circle?” (The set of points equidistant from the center.) “Why do both of these methods work?” If no one used one of the methods, bring it up and ask students if it would work. (Points that work in the equation must have a distance of 5.9 from \(P\). Points that have a distance of 5.9 from \(P\) are on the circle by definition.)

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).

Reference the diagrams from the previous activities. The point of intersection of the altitudes was \(H=(2,1.2)\) and the point of intersection of the perpendicular bisectors was \(P=(4,4.4)\).

Invite students to share their strategies for proving the 3 points are collinear. Then tell students, “This is called the Euler Line. This happens in all triangles, not just this one.”

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.

Tell students that tessellations are infinite but they need not spend the entire day drawing the first tessellation in this activity. Folding graph paper in half gives 4 sections to work in (front and back). Instruct students to consider half the page their “plane.”

If students use horizontal and vertical lines for the rectangles, tell them to make right triangles without using either horizontal or vertical lines.

If students struggle to find a third shape that tiles the plane, suggest they consider equilateral triangles or regular hexagons.

Invite several students to share their equations for the right triangle. Ask the class how they could check if these sets of equations outline right triangles. (Graph them or verify that a pair of slopes has a product of -1.)