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Prove or explain why three reflections can transform a figure onto another figure if they are congruent

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Prove or explain why three reflections can transform a figure onto another figure if they are congruent

Dec 16, 2016

#1
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Hi Mellie,

It is always great to see you on the forum :)

Prove or explain why three reflections can transform a figure onto another figure if they are congruent

I don't have a great explanation here but maybe I can do so pictures to help you think one up for yourself - or someone else might weigh it.

I just realized that this is only 2 reflections.  Did it have to be three or could it be 3 or less??

Never mind, I have already inserted the first triangle pic so I might as well talk about it.

1)  I started with the brown triangle on the right and wanted to reflect it across to the brown triangle on the left.

2)  I drew a perpendicular bisector between two corresponding vertices A and A'.

3)  Then I reflected the triangle on the right across this line, resulting in the yellow triangle.

4)  Then I drew another perpendicular bisector between 2 corresponding vertices. C' and C''.  This will always go through A.

5)  Then I reflected the yellow triangle across this line and that puts me in the position of the initial second triangle

Ok, now I will go think, and play, some more. :D

Dec 16, 2016
#2
+118608
+2

Here I have done exactly the same thing again with an irregular hexagon.

It still only took two reflections

Dec 16, 2016
#3
+118608
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Here I have done another, I used the same method but this time it did take 3 reflections.

This is why it is different from the others.

FFor the first two you could get from the forst shape to the second shap using translation and rotation.

The corresponding points A,b,c etc were both a clockwise direction.

With the example below, a reflection was also involved,  in the initial pentagon on the right the points go clockwise but the congruent shape on the left (the brown one) the letters go in a anticlockwise direction.

Dec 16, 2016