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[/img][/td][td][img]data:image/png;base64,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[/img][/td][/tr][/table]
[size=150]Prove that if a point [math]C[/math] is the same distance from [math]A[/math] as it is from [math]B[/math], then [math][/math]C must be on the perpendicular bisector of [math]AB[/math].[br]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 [math]C[/math] is on, and prove that line has to be the perpendicular bisector.”[br][br]They each drew a line and thought about their pictures. Here are their rough drafts.[/size][br][br][table][tr][td][br][br]Diego’s approach:[br] “I drew a line through [math]C[/math] that was perpendicular to [math]AB[/math] and [br]through the midpoint of [math]AB[/math]. [br]That line is the perpendicular bisector of [math]AB[/math] and [math]C[/math] is on it,[br] so that proves [math]C[/math] is on the perpendicular bisector.”[/td][td][img]data:image/png;base64,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[/img][/td][/tr][/table][table][tr][td]Jada’s approach:[br] “I thought the line through [math]C[/math] would probably go through [br]the midpoint of [math]AB[/math] so I drew that and labeled the midpoint [math]D[/math]. [br]Triangle [math]ACB[/math] is isosceles, so angles [math]A[/math] and [math]B[/math] are congruent, [br]and [math]AC[/math] and [math]BC[/math] are congruent. And [math]AD[/math] and [math]BD[/math] are congruent [br]because [math]D[/math] is a midpoint. [br]That made two congruent triangles by the Side-Angle-Side [br]Triangle Congruence Theorem. So I know angle [math]ADC[/math] and [br]angle [math]BDC[/math] are congruent, but I still don’t know [br]if [math]DC[/math] is the perpendicular bisector of [math]AB[/math]."[/td][td][img]data:image/png;base64,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[/img][/td][/tr][/table][table][tr][td]Noah’s approach: [br]“In the Isosceles Triangle Theorem proof, [br]Mai and Kiran drew an angle bisector in their isosceles triangle, [br]so I’ll try that. I’ll draw the angle bisector of angle [math]ACB[/math]. [br]The point where the angle bisector hits [math]AB[/math] will be [math]D[/math]. [br]So triangles [math]ACD[/math] and [math]BCD[/math] are congruent, [br]which means [math]AD[/math] and [math]BD[/math] are congruent, [br]so [math]D[/math] is a midpoint and [math]CD[/math] is the perpendicular bisector."[/td][td][img]data:image/png;base64,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[/img][/td][/tr][/table][br][size=150]With your partner, discuss each student’s approach.[/size][list][*]What do you notice that this student understands about the problem?[/*][/list]
[list][*]What question would you ask them to help them move forward?[/*][/list]
Using the ideas you heard and the ways you think each student could make their explanation better, write your own explanation for why [math]C[/math] must be on the perpendicular bisector of [math]A[/math] and [math]B[/math].[br]
[size=150]Elena has another approach: “I drew the line of reflection. If you reflect across C, then A and B will switch places, meaning A' coincides with B, and B' coincides with C. will stay in its place, so the triangles will be congruent.”[/size][br][br]What feedback would you give Elena?
Write your own explanation based on Elena‘s idea.
If [math]P[/math] is a point on the perpendicular bisector of [math]AB[/math], prove that the distance from [math]P[/math] to [math]A[/math] is the same as the distance from [math]P[/math] to [math]B[/math].
[list][*]What do you notice that your partner understands about the problem?[/*][/list]
[list][*]What question would you ask them to help them move forward?[/*][/list]