An Algebraic Look at the Golden Ratio

An Algebraic Look at the Golden RatioName______

Period_____Date______

You might remember from geometry that the golden rectangle

was special because if we cut a square from it,

we still had a rectangle that was similar to the

original rectangle. Let’s look at the algebra

behind this: x

x y

According to the definition of a golden rectangle, what is true about the two rectangles in the figure?

According to the definition of similar figures, what proportion must be true in this figure?

Let y = 1, and show work to solve for x. Write your answer in both exact and decimal forms.

Let x = 1 and show work to solve for y. Write your answer in both exact and decimal forms.

Show that one of these numbers is the reciprocal of the other.

What else do you notice about these two numbers?

The first (larger) number that you got is called phi () and is the golden ratio.

Look back at a construction of the golden rectangle.

Let AC = x. Show work to find the lengths of each of the following segments, in terms of x.

AB = ______

AE = ______

EC = ______

EF = ______

AF = ______

BF = ______

The golden ratio should be the ratio of long side to short side in either (actually both) of the rectangles.

Find and simplify.

Find and simplify.

The golden triangle is an isosceles triangle such that the ratio of the lengths of a leg to the base is equal to the golden ratio.

Sketch a golden triangle in this golden rectangle. (There are 4 such triangles. Can you see all of them? Their vertices don’t lie on the rectangle!)

Draw the altitude of your triangle. Use your knowledge of right triangle trigonometry to find the measure of a base angle of this golden triangle.

What is the measure of the interior angle of a regular pentagon?

What would be the measure of an exterior angle of

this regular pentagon?

If the sides of this regular pentagon were extended, what would be the result?

Use your calculator to approximate this continuing fraction:

What do you think is the actual value of this continuing fraction?

Use algebra to prove it. (Hint: :Let x = 1 + and realize that the same expression is in the denominator…)

Use your calculator to approximate this infinite radical:

What do you think is the actual value of this infinite radical?

Use algebra to prove it using the same substitution idea.