2013 Mathcounts State Prep: Similar Triangles and Height to the Hypotenuse

There are many concepts you can learn from this image, which cover numerous similar right triangles, ratio/proportion/dimensional change and the height to the hypotenuse.

Question:
#1: Δ ABC is a 3-4-5 right triangle. What is the height to the hypotenuse? 
Solution: 
Use the area of a triangle to get the height to the hypotenuse. 
Let the height to the hypotenuse be "h"
The area of Δ ABC is  $\frac{3\ast 4}{2}$= $\frac{5\ast h}{2}$
Both sides times 2 and consolidate: h =  $\frac{3\ast 4}{5}$ = $\frac{12}{5}$

Practice: What is the height to the hypotenuse?

Question:
#2: How many similar triangles can you spot?
Solution: 
There are 4 and most students have difficulty comparing the largest one with the other smaller ones.
Δ ABC is similar to Δ ADE, Δ FBD, ΔGEC. Make sure you really understand this and can apply this to numerous similar triangle questions. 

Question: 
#3: What is the area of  □ DEGF if $\overline{BF}$ = 9 and $\overline{GC}$ = 4
Solution: 
Using the two similar triangles Δ FBD and  ΔGEC (I found using symbols to find the corresponding legs
 to be much easier than using the lines.), you have $\frac{\overline{BF}}{\overline{FD}}$ = $\frac{\overline{GE}}{\overline{GC}}$.
s (side length of the square) = $\overline{GE}$ =  $\overline{FD}$
Plug in the given and you have 9 * 4 = s² so the area of □ DEGF is 36 square units. (each side then is square root of 36 or 6)

Question: 
#4: Δ ABC is a 9-12-15 right triangle. What is the side length of the square? 
Solution :
The height to the hypotenuse is$\frac{9\ast 12}{15}$ = $\frac{36}{5}$
Δ ABC is similar to Δ ADE. Using base and height similarities, you have $\frac{\overline{BC}}{\overline{DE}}$ = $\frac{15}{S}$ = $\frac{\frac{36}{5}}{\frac{36}{5}-S}$
Cross multiply and you have 108 - 15*S = $\frac{36}{5}$ *S
S =$\frac{180}{37}$