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The three perpendicular bisectors of a triangle always intersect in one point. We can use coordinate geometry to show that the altitudes of a triangle intersect in one point, too. The three altitudes of triangle
The slopes of sides
We can now verify that
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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
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).
Triangle
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 because the line segments are perpendicular.)
If students struggle to verify algebraically that
As in the previous activity, an index card can be a useful tool 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 because the line segments are perpendicular.) Invite students to share strategies for verifying their coordinates of
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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.)
Triangle
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
Graph paper
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.
A tessellation covers the entire plane with shapes that do not overlap or leave gaps.
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.)