Assignment 4B (Due Monday, 10/29)

This problem is to write up the problem left at the end of the hour Wednesday 10/23, since it is a very good problem for figuring out how to get ratios from information about shapes. I have put in labels this time so that I can be sure I am being clear about which sides are equal.

Then there are hints below that break the problem down into steps. I suggest trying the problem for a while, then if you get stuck try the hints. (While the pentagon is motivation, everything you need to know to solve this problem is in the "givens" below. (If you did it without the hint, you are welcome to tell us.)

4.2 A special triangle (10 points)

In this triangle AC = AB = d and CB = CD = DA = s.

MAJOR PROBLEM: Let the side of a regular pentagon be s and the length of a diagonal be d, use this figure to find the ratio d/s.  (Hint:  Is it legit to set s = 1 to simplify?)

Hint 1: Partial solution

Let ABC be an isosceles triangle (AB = AC). Suppose there is a point D on segment AB so that CD = CB. Let us label the lengths: AB = AC = x and CD = CB = y.

• With these assumptions what is length BD in terms of x and y?
• Also, write the length DA in terms of x and y.

Hint 2: Rest of the solution

If we make the additional assumption that triangle DAC is also isosceles (i.e., DA = DC = y), find an equation in x and y that must be true.

• Then solve this equation to find the ratio K = x/y. If you don't see how to do this, start by assuming y = 1 and find x = x/1. This should give x as an exact real number.
• Then see if you can figure out what the ratio is in general (it had better be the same; can you justify this).