Perpendicular Bisectors in Triangles
Materials Needed: Patty paper, ruler, protractor, colored pencils
Lesson Opener Activity
1. On a sheet of patty paper, draw and label .
2. Fold the paper so that A lays directly on top of B. Crease the fold.
3. Open the paper and draw a line on the crease from edge to edge.
4. Label the point of intersection between and the folded line M.
5. Label a second point anywhere on the creased line as C.
6. Measure and. MA = ______MB = ______
7. Measure CMA and CMB. mCMA = ______mCMB = ______
8. Based on your measurements, use correct vocabulary to describe the pair of intersecting lines on your paper?
______
9. Draw a segment from C to A, then another segment from C to B.
10. Measure and CA = ______CB = ______
11. Label another point on the creased line as D.
12. Repeat steps 9 and 10. DA = ______DB = ______
13. Based on your observations, make a conjecture about any point on the perpendicular bisector of a segment. ______
Materials Needed: Patty paper, ruler, protractor, colored pencils, compass
Group Activity (groups of three or four)
1. One student in the group will draw a large acute triangle, one will draw a large right triangle, and one will draw a large obtuse triangle on a piece of patty paper. The fourth student can draw a triangle of his/her choice. Label it ∆ABC.
2. As in the previous activity, fold one side of the triangle to form the perpendicular bisector of the side (fold your paper so that A lies directly on top of B and crease the line).
3. Repeat for the other two sides. What do you notice about the three creased lines? ______
4. Label the point of intersection of the creased lines D. The point of intersection of three or more lines is also called the point of concurrency.
5. Using a colored pencil, draw a line on one crease from the side of the triangle to point D. Repeat for the other two sides. Verify that the lines are perpendicular and the sides of the triangle are bisected and label the diagram showing this.
6. Using a different colored pencil, draw a dashed line from point D to each vertex of the triangle.
7. Measure the length from each vertex to point D.
DA = ______DB = ______DC = ______
8. Compare your results with the other members of your group. What can you conclude about the point of intersection of the perpendicular bisectors of a triangle?
______How is this conjecture related to the Opening Activity? ______
9. Compare your triangle with the same type in other groups. Describe the location of the point of intersection of the perpendicular bisectors on all three triangles.
Acute Triangle ______
Right Triangle ______
Obtuse Triangle ______
Extension Activity
Using a compass, construct a circle with the center at D and the radius the length of Compare the circle you constructed with your group members. What do you notice about the relationship between the triangle and the circle you constructed? ______
v The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter because it is the center of the circle that is circumscribed about (goes around the outside of) the triangle.