In the right triangle below, altitude is drawn.
a) State all of the triangles in the figure that are similar.
b) If AB = 2 and AD = 1, find all of the edge lengths in the figure.
c) Find the scale factor that relates each of the triangles. This exercise has three solution values.

Problem 2)

If a < b are any two positive numbers, is it always true that the geometric mean is in-between them? That is to say, is it true that ?

Problem 3)

You are in a forest with a map and find that you have just walked from one point to another, where the second point is 1 mile east and 3 miles north of where you started. How far have you walked?

Problem 4)

Solve for the unknown. Notice that each is a kind of relationship that could appear in a geometric mean.
a)
b)
c)
d)
e)
In each of (a) through (e) here, there is a right triangle with segment lengths that stand in these relationships. Draw and label a triangle for each of these.

Problem 5)

Find the area of an equilateral triangle with edge length 10.

Problem 6)

If 2x+5 : x+10 = 2:3 solve for x.

Problem 7)

There is a famous quantity, like , called which is pronouned "phi". It is also known as the "Golden Ratio". To find it, do the following.
Suppose you have a line segment with point C drawn in-between A and B. C is located so that the ratio AB:AC = AC:CB. If we let AB = 1 then AC = . Try to solve for it.
Hint: What you really need to solve is . Make sure you understand why this equation is correct.

Problem 8)

a) Using the diagram from problem 1, prove that . Hint: This uses the similarity between .
b) Prove that .
c) Use the facts that together with the fact that to prove that .

Problem 9)

I'm not sure if you've seen this kind of material before, but
a) if the sides of a triangle are 3, 4, and 6, is it right, acute, or obtuse?
b) if the sides are 10, 11, and 12?