by: Doris Santarone
Final Assignment #2: Square Inscribed In A Semicircle
Part 1:
Given a semicircle with an inscribed square of side s.
First, I had to construct this. I started with a semicircle (which is constructed in GSP as an arc through 3 points, with the two of those points as endpoints of a diameter of the circle.)
Next, I constructed a square where one side is the diameter of the circle.
Then, I constructed the line through the vertices of the square and the center of the circle. Then, I found the intersection points of these diagonals with the semicircle. These points will be two of the vertices of my inscribed square.
Lastly, I used these two points to construct my inscribed square.
Part 2:
Given a semicircle with an inscribed square of side s. Let a be the length on the diameter on each side of the square. (Click HERE to get the GSP Sketch below.)
We want to find 
.
Since the radius of the circle is half the diameter, then the radius of the circle is: 
.
Now, I will construct a radius of the circle from the center to one of the vertices of the square that is on the semicircle. This forms a right triangle
Using the Pythagorean Theorem to start, then simplifying using algebra (and factoring using the quadratic formula), we get:
It's PHI!!!!! Wow!!! (We can eliminate the - because we are dealing with a ratio of lengths, which will always be a positive number).
Click HERE to see the Geometer's Sketchpad sketch of this problem. Press the "Change the Size of the Semicircle" button to make the semicircle larger and smaller. Notice that a and s change size, but remains equal to 
.