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 3∗42= 5∗h2
Both sides times 2 and consolidate: h = 3∗45 = 125
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 ¯BF = 9 and ¯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 ¯BF¯FD = ¯GE¯GC.
s (side length of the square) = ¯GE = ¯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 is9∗1215 = 365
Δ ABC is similar to Δ ADE. Using base and height similarities, you have ¯BC¯DE = 15S = 365365−S
Cross multiply and you have 108 - 15*S = 365 *S
S =18037
