Exam 6 Exercises

Author:: Adam

Problem 1)

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)

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?