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ΔABC is a triangle with no equal sides. A new triangle congruent to ΔABC is to be placed on the plane such that the new triangle shares exactly two vertices with ΔABC. At how many different locations can the new triangle be placed?

Sep 25, 2018

#1
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I believe that  "3"  is the answer

Sep 25, 2018
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Sep 25, 2018
#3
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My initial instinct was 3 as well; this visual may help you. I have found three ways to orient the two congruent scalene triangles such that two both triangles share both vertices. Does this help you at all?

Sep 26, 2018
edited by TheXSquaredFactor  Sep 26, 2018
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Sep 26, 2018
#5
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I was trying to guide you in the right direction. I think you misinterpreted what I was attempting. When I said that there are three ways to orient the two congruent triangles such that they both share exactly two vertices, I had intentionally excluded some for you to figure out yourself.

Notice how those orientations only concern two vertices in particular. Notice that there are two other pairs of vertices that we have not yet considered. Using the logic from before, if there are three ways for two vertices, this would also be true for two remaining pairs of vertices as well.

$$3*3=9 \text{ different locations}$$

TheXSquaredFactor  Sep 26, 2018