## Friday, December 21, 2012

### 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  $$\Large\frac{3*4}{2}$$= $$\Large\frac{5*h}{2}$$
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 =$$\frac{\Large{180}}{\Large{37}}$$