Diego, Jada, and Noah were given the following task:[br]Prove that if a point [i]C [/i]is the same distance from [i]A [/i]as it is from [i]B[/i], then must be on the[br]perpendicular bisector of [i]AB[/i].

At first they were really stuck. Noah asked, “How do you prove a point is on a line?” Their[br]teacher gave them the hint, “Another way to think about it is to draw a line that you know [i]C[/i][br]is on, and prove that line has to be the perpendicular bisector.”

Diego’s approach: “I drew a line through [i]C[/i] that was[br]perpendicular to [i]AB [/i]and through the midpoint of [i]AB[/i]. That[br]line is the perpendicular bisector of [i]AB [/i]and is on it, so[br]that proves [i]C[/i] is on the perpendicular bisector.”

Jada’s approach: “I thought the line through [i]C[/i] would[br]probably go through the midpoint of [i]AB[/i] so I drew that and[br]labeled the midpoint [i]D[/i]. Triangle [i]ACB[/i] is isosceles, so angles [i]A[/i][br]and [i]B [/i]are congruent, and [i]AC[/i] and [i]BC [/i]are congruent. And [i]AD[/i][br]and [i]DB[/i] are congruent because [i]D[/i] is a midpoint. That[br]made two congruent triangles by the Side-Angle-Side[br]Triangle Congruence Theorem. So I know angle [i]ADC[/i] and [br]angle [i]BDC [/i]are congruent, but I still don’t know if [i]DC [/i]is the[br]perpendicular bisector of [i]AB[/i].”

Noah’s approach: “In the Isosceles Triangle Theorem proof,[br]Mai and Kiran drew an angle bisector in their isosceles[br]triangle, so I’ll try that. I’ll draw the angle bisector of angle [i]ACB[/i][br]. The point where the angle bisector hits will be [i]D[/i].[br]So triangles [i]ACD[/i] and [i]BCD [/i]are congruent, which means [i]AD[/i][br]and [i]BD[/i] are congruent, so [i]D [/i]is a midpoint and [i]CD[/i] is[br]the perpendicular bisector.”

Triangle [i]ABC[/i] is isosceles. [br]Mark congruent sides and angles with the pen tool. [br]Use the construction tools to create the perpendicular bisector of [i]AB.[/i]

IM G Unit 2 Lesson 8 from IM Geometry by Illustrative Mathematics, [url=https://im.kendallhunt.com/HS/students/2/2/1/index.html]https://im.kendallhunt.com/HS/students/2/2/8/index.html[/url]. Licensed under the Creative Commons Attribution 4.0 license, [url=https://creativecommons.org/licenses/by/4.0/]https://creativecommons.org/licenses/by/4.0/[/url].