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 \(\Large\frac{3*4}{2}\)= \(\Large\frac{5*h}{2}\)
Both sides times 2 and consolidate: h = \(\Large\frac{3*4}{5}\) = \(\Large\frac{12}{5}\)
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 \(\frac{\Large{\overline{BF}}}{\Large{\overline{FD}}}\) = \(\frac{\Large{\overline{GE}}}{\Large{\overline{GC}}}\).
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\(\frac{\Large{9*12}}{\Large{15}}\) = \(\frac{\Large{36}}{\Large{5}}\)
Δ ABC is similar to Δ ADE. Using base and height similarities, you have \(\frac{\Large{\overline{BC}}}{\Large{\overline{DE}}}\) = \(\frac{\Large{15}}{\Large{S}}\) = \(\frac{\frac{\Large{36}}{\Large{5}}}{\frac{\Large{36}}{\Large{5}} - \Large{S}}\)
Cross multiply and you have 108 - 15*S = \(\frac{\Large{36}}{\Large{5}}\) *S
S =\(\frac{\Large{180}}{\Large{37}}\)
