1. triangle ABC and triangle DEF are not similar, and they satisfy the following conditions:
• AB > DE
• BC > EF
• AC > DF
Grogg claims that triangle ABC must have a greater area than triangle DEF does. Do you agree with Grogg? If yes, prove his claim. If no, provide a counterexample to Grogg’s claim.
2. As shown in the diagram, X,Y, and Z are the midpoints of the sides of equilateral triangle ABC with center O. Describe a sequence of transformations that maps triangle ABC to triangle XYZ. (In particular, you must map A to X, B to Y, and C to Z.) You must describe each transformation precisely. For example, if you use a reflection, you must specify the line you are reflecting over.
3. In triangle ABC, sin A : sin B : sinC = 2 : 3 : 4. Find cos(A + C).
4. The combined area of all the faces of a rectangular prism is 12, and the combined length of all its edges is 19. Find the length of a space diagonal of the prism.
Numbers 3 and 4 have been answered here: