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how can i prove that two figures are similar?

Hi,
I have two figures in a graph and i have to prove that the figures are similar.They both look the same but one of them is smaller.I know that in order to prove that this two figueres are similar the angles must be congruent so i am using the slope formula to prove that they are similar.
My question is Does every slope of corresponding sides have to be equal in order for two figures to be similar?
 
 
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2 Answers

Daisy, you can use SSS similarity.  If you can prove the ratios of all three of the corresponding sides are equal Then the triangles must be similar.  You can use the distance formula to compute the lengths.
Hi Daisy;
One figure could be a rotation, reflection or translation of the other.
 
If one figure is a rotation of the other, then the three slopes would likely be different.
 
If one figure is a reflection of the other, the slopes would be negatively identical.
 
If one figure is a translation of the other, the three slopes would be identical.
 
Please comment on whether the two figures are rotations, reflections or translations of the other. 
 
 
 

Comments

The short answer is:  YES (!!), the CORRESPONDING angles (an equivalent for the slope) MUST be equal ... as long as any pairing of angles from one triangle to that of another triangle, without duplication, provides equivalency, the triangles are similar.
 
However, "similar" can be determined with only two angles/slopes, AA ... does not require AAA. this is obvious, because the (three) angles in a triangle must add up to 180 degrees (2 pi radians); hence if two of the angle pairings are congruent, the third angle pairing must, by definition, be congruent.
 
Rotation. reflection, translation are irrelevant ... there is absolutely no need to determine any of these transformations ... only the equivalence of two parings of angles.
For example let's say that triangle XYZ is a mirror image, reflected on the y-axis, of triangle ABC; and we're dealing with scalene triangles (all angles unequal, only for clarity purposes).  Then, A corresponds to Z, B corresponds to Y, and C corresponds to X; side AB corresponds to YZ, side AC corresponds to ZX, and BC corresponds to YX.  Since the question stated "corresponding sides" (which also requires "corresponding angles"), transformation is moot.