Are all triangles in the universe similar to 3-4-5?

(a) AB = 8, BC = 6 in rectangle ABCD. Find the lengths of all segments shown in the diagram above.
Specifically: BD, CF,DE, BE, FE, CE

Comments: This is another in a series of rectangle investigations. To deepen student understanding of triangle relationships and to provide considerable practice with these ideas, the question asks for more than just one result. Students should be encouraged to first draw ALL of the triangles in the diagram separately and recognize why they are all similar! Using ratios, students should be able to find all the segments efficiently. One could also demonstrate the altitude on hypotenuse theorems as well!

(b) In case, students need a bit more of a challenge, have them derive expressions for all of the above segments given that AB = b and BC = a. To make life easier, assume b > a. This should keep your stronger students rather busy! This algebraic connection is powerful stuff. We want our students to appreciate that algebra is the language of generalization.