Notes for the Warm-Up: It was given (before everything was written on!) that BE and CD were parallel, which gave us congruent corresponding angles (<ABE, and <C) marked with 1 arc mark each. We also note that <A is the same in the smaller and bigger triangles, so we say it is congruent to itself by the reflexive property, and mark it with 2 arc marks. This allows us to say that the two triangles are similar by AA.
For part b: Initially, most students set up ratios of 15 to 18, and 10 to 12 (or something equivalent to those), because those sides correspond with one another. Remember that two pairs of proportional sides is not enough to conclude triangles are similar! We then note that side RT is a shared side, and so is congruent to itself in both triangles. When we make that 3rd ratio of corresponding sides (RT to RT), that would be 1, which is not equal to 10/12 or 15/18. So the corresponding sides are not proportional, so the triangles are not similar!
For part c: Be careful when setting up proportions from diagrams like the one in part a. The lengths of BE and CD are whole side lengths in their respective triangles (BE in the smaller triangle, CD in the bigger triangle). So if we compare BE to CD (since they are corresponding sides in their two triangles, they both “function” the same way), we need to make sure the other ratio is a whole side in the smaller triangle, to a corresponding whole side in the bigger triangle, which would be 8 to 12, NOT 8 to 4!. This gives us x/18=8/12, and then BE=12.
We then reviewed some problems on the green Simplifying Radicals WS, and the yellow Geometric Mean WS. See below for ideas covered for approaching Geometric Mean problems:
We also then reviewed 3 problems on p.268: 11: a-c, practice with the triangle proportionality theorem. Key idea: Make sure your parts in your ratios correspond, especially when you are dealing with cross section sides (like in the warm-up)!