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Right Triangle Similarity Surprising Results
XYZ , XYA, and YAZ are all right triangles
1.) Suppose mX  40 . Find the measures of all of the
other angles:
mY  _____
m2  _____
mA  _____
mZ  _____
m1  _____
2.) Suppose mX  20 . Find the measures of all of the other angles:
mY  _____
m2  _____
mA  _____
mZ  _____
m1  _____
3.) Suppose mX  a
mY  _____
mA  _____
m1  _____
m2  _____
mZ  _____
4.) You have enough information to show that XYZ ~ XAY ~ YAZ . How could you do this?
5.) The following triangles are similar.
long _ leg 2
long _ leg 1
XA XY

or
.

XY XZ
hypotenuse2 hypotenuse1
long _ leg 3
long _ leg 2
XA AY

Because XYA ~ YZA , by the def. of similarity,
or
.

AY AZ
short _ leg 2 short _ leg 3
Because XZY ~ XYA , by the def. of similarity,
Substitute the lengths of the sides:
d a
d e
 and 
a
e
6.) Using the ideas from #5, write the lengths on the original diagram, and write all of the equal
ratios of sides you can find.
7.) Using similar triangles, solve for x:
8.) For what value of x is this figure possible?
9.) The geometric mean of two numbers a & b is x, where
the geometric mean of two others?
10.) What is the geometric mean of 4 & 9?
11.) Solve for x:
a x
 . In #7, which side length is
x b
Cool fact #1: Similarity of right triangles can lead to another proof of the Pythagorean Theorem.
Why is it true that:
c a
c b
 and  ?
a x
b y
What is the result of cross-multiplication?
Add the two equations: a 2  b 2 
Substituting, as you know x+y=c:
Which is what we wanted!
Cool fact #2: Given a segment of length k and a unit segment 1, how can you construct a
segment of length k ?
A prerequisite fact is that any interior angle of a circle that spans the diameter is always right.
By June we will know how to prove this. For now, we will just apply this fact.
Add segments k and 1.
Then, construct a circle with k+1 as the diameter. How can you do this?
Think of how you would first find the radius.
Then, construct a perpendicular line through the endpoints of the two
segments (as shown to the right). The altitude of the triangle has length k . Why?