So there are at least five similarity transformations, but these clearly aren't all of them, since the composition of a dilation and certain types of isometries is neither an isometry nor a dilation.Īs it turns out, the composite of a dilation and a translation is another dilation, only with a different center but the same scale factor as the original dilation. Now a similarity transformation is the composite of a dilation and an isometry. We know that for congruence, there are only four isometries - reflections, rotations, translations, and glide reflections. Here are a few more things that I want to say about similarity. Recall that in the U of Chicago, there's no Division (or Subtraction) Property of Equality, so we should think of deriving PT/ PS = 1/2 by multiplying by 1/ PS, not dividing by PS. ![]() Given: T is the midpoint of PS Q is the midpoint of PR.Ģ. The text gives one example of a proof using SAS Similarity, written in two-column format: Then one can prove A'B'C' and XYZ congruent by SAS Congruence, just as in the Wu and U of Chicago proofs. First B' is chosen on AB so that A'B' = XY (Ruler Postulate), and then C is chosen so that B'C' | | BC (Playfair), and then the Corresponding Angles Consequence gives enough congruent angles to conclude ABC ~ A'B'C' by AA. It appears what these are doing is proving SAS Similarity using AA Similarity in lieu of dilations. ![]() This makes the proof easier to understand, as A is actually the same point as A' - so Angles A and A' are congruent by the Reflexive Property.Īs I mentioned before, I've seen pre-Common Core texts that give a proof of the SAS Similarity Theorem, yet declare AA to be a p ostulate. Notice that neither Wu nor the U of Chicago choose any arbitrary dilation with the correct scale factor - both choose the dilation centered at exactly the point A with the correct scale factor. Thus Triangle ABC can be mapped onto XYZ by a composite of dilations and reflections, so Triangle ABC ~ XYZ. So Triangle A'B'C' is congruent to XYZ by the SAS Congruence Theorem. Let k = XY/ AB, and then find Triangle A'B'C' ~ ABC with scale factor k. Given: Triangles ABC and XYZ with Angle A = X, AB/ XY, AC/ XZ. Since 11 is an odd number, this question is included with the answers in the back of the book. The U of Chicago text directs the students to prove SAS Similarity in Question 11, and I will likewise include it on my worksheet an an exercise for students to complete. I won't bother posting the Wu proof, as it's nearly identical to this one, except with different notation (in particular, Wu uses A'B'C', not XYZ, to denote the triangle to be proved similar to ABC, and so he comes up with notation like A*B*C* or A0B0C0 to denote the dilation image of ABC). As it turns out, the center of the dilation is irrelevant - any dilation with the correct scale factor will work. The above proof finds exactly such a dilation - namely one with scale factor XY/ AB. Notice that in some ways, Wu's definition of similar works better here - we want to show that there exists some dilation D such that D( ABC) is congruent to Triangle XYZ. Thus Triangle ABC can be mapped onto XYZ by a composite of dilations and reflections. So Triangle A'B'C' is congruent to XYZ by the ASA Congruence Theorem. With transitivity, Angle A' = X and Angle B' = Y. Also, since dilations preserve angle measure, Angle A = A' and Angle B = B'. Since XY and AB are corresponding sides, let k = XY/ AB be the magnitude of a dilation applied to Triangle ABC. Given: Triangles ABC and XYZ with Angle A = X and Angle B = Y. ![]() If two triangles have two angles of one congruent to two angles of the other, then the triangles are similar. I'll keep this one in paragraph form for now, as it is in the text, but as usual, I made cosmetic changes such as replacing "size transformation" with "dilation": Once again, this is in stark contrast to most texts where AA is a postulate. So let's take a look at the proof of AA Similarity as given in the U of Chicago text. This section gives a proof of AA and asks the students to prove SAS. For he states the theorems in the reverse order from the U of Chicago - in the text SSS is first, then AA, then SAS, while for Wu, SAS is Theorem 26, AA is Theorem 27, and only afterward does SSS appear. The reason that I cover these two first and skip over SSS Similarity is because of Dr. Section 12-9 of the U of Chicago text covers the AA and SAS Similarity Theorems.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |