Mathematics: Triangles similar to the same triangle are to each other. a. congruentb) similarc) closed) parallel

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Triangles similar to the same triangle are to each

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lexus1427

12.03.2022

Similar.

First of all, options C and D make no sense, since we don't have a concept for triangles being "close" or "parallel" to each other.

Now, two triangles are similar if one is the scaled version of each other. As a consequence, the triangles have the same angles, and the correspondant sides are in the same proportion.

So, assume that we start from the original triangle ABC. We build two other triangles, DEF and GHI, both similar to ABC.

This means that the correspondant sides of ABC and DEF are in the same proportion, say . This means that

$DE = r\cdot AB,\quad EF = r\cdot BC,\quad DF = r\cdot AC$

Similarly, the correspondant sides of ABC and GHI are in the same proportion, say . This means that

$GH = s\cdot AB,\quad HI = s\cdot BC,\quad GI = s\cdot AC$

Now, the question is: are DEF and GHI similar? The answer is yes, because we can perform the following transformation: let's start for example from the side GH:

$GH \to \cfrac{1}{s} GH = AB \to AB \cdot r = DE$

This means that

$GH \cdot \cfrac{r}{s} = DE$

and thus the sides of DEF and GHI are in the same proportion, which means that they are similar.

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Triangles similar to the same triangle are to each other. a. congruentb) similarc) closed) parallel

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