IM Geometry Unit 2 Lesson 8
Which one doesn't belong?
Diego, Jada, and Noah were given the following task:
Prove that if a point C is the same distance from A as it is from B, then must be on the
perpendicular bisector of AB.
At first they were really stuck. Noah asked, “How do you prove a point is on a line?” Their
teacher gave them the hint, “Another way to think about it is to draw a line that you know C
is on, and prove that line has to be the perpendicular bisector.”
Diego’s approach: “I drew a line through C that was
perpendicular to AB and through the midpoint of AB. That
line is the perpendicular bisector of AB and is on it, so
that proves C is on the perpendicular bisector.”
Jada’s approach: “I thought the line through C would
probably go through the midpoint of AB so I drew that and
labeled the midpoint D. Triangle ACB is isosceles, so angles A
and B are congruent, and AC and BC are congruent. And AD
and DB are congruent because D is a midpoint. That
made two congruent triangles by the Side-Angle-Side
Triangle Congruence Theorem. So I know angle ADC and
angle BDC are congruent, but I still don’t know if DC is the
perpendicular bisector of AB.”
Noah’s approach: “In the Isosceles Triangle Theorem proof,
Mai and Kiran drew an angle bisector in their isosceles
triangle, so I’ll try that. I’ll draw the angle bisector of angle ACB
. The point where the angle bisector hits will be D.
So triangles ACD and BCD are congruent, which means AD
and BD are congruent, so D is a midpoint and CD is
the perpendicular bisector.”
Triangle ABC is isosceles.
Mark congruent sides and angles with the pen tool.
Use the construction tools to create the perpendicular bisector of AB.
IM G Unit 2 Lesson 8 from IM Geometry by Illustrative Mathematics, https://im.kendallhunt.com/HS/students/2/2/8/index.html. Licensed under the Creative Commons Attribution 4.0 license, https://creativecommons.org/licenses/by/4.0/.