Illustrate each of these definitions with a sketch using four different triangles. Try to draw four triangles that are different in significant ways -- different side lengths, angles, and types of triangles. The first one in definition (a) is done as an example.
A triangle has three altitudes, one from each vertex. (An altitude of a triangle is a line segment connecting a vertex to the line containing the opposite side and perpendicular to that side.)
A triangle has three medians. (A median is a segment connecting any vertex to the midpoint of the opposite side.)
A triangle has three midlines. (A midline connects two consecutive midpoints.)
Draw five triangles, each on its own piece of patty paper. Use one triangle for each construction below.
Carefully construct the three altitudes of the first triangle.
Carefully construct the three medians of the second triangle.
Carefully construct the three midlines of the third triangle.
Carefully construct the three perpendicular bisectors of the fourth triangle.
Carefully construct the three angle bisectors of the fifth triangle.
When you construct medians, you need to do two things: First find the midpoint; then fold or draw a segment connecting that point to the opposite vertex. Except in the case of special triangles (such as an equilateral triangle, and one median in an isosceles triangle), you can't construct a median with just one fold. When you construct altitudes, you need to construct a perpendicular to a segment, but not necessarily at the midpoint of that segment. And remember that the altitude may fall outside the triangle, so you might want to draw or fold an extension of the sides of the triangle to help you. Close Tip
Video Segment In this video segment, participants construct the altitudes, medians, and midlines of their triangles. Compare your solutions to Problem C4 with those in this video segment.
What are the similarities and differences in your results? What conjectures can you make about the constructions you've just completed?
If you are using a VCR, you can find this segment on the session video approximately 15 minutes and 40 seconds after the Annenberg Media logo.