Find a Ratio
By: Erica Fletcher
Part 2 of Final: Investigation 2
Below I constructed a square with side s that is inscribed in a semicircle. The length a is on each side of the square along the diameter. If you would like to construct a figure like the one shown below use this GSP file.
Goal: For this assignment we want to find the ratio of s to a.
In order to begin this construction we need to first construct a square with side lengths s. Then after constructing the square we first had to construct the midpoint of any side of the square. This midpoint is connected to the opposite vertex of the square. Now using this midpoint and segment we can construct a circle by center + radius. Lastly, we constructed a semicircle by constructing an arc using 3 points.
We can see that the midpoint of the side of the square along the diameter is also the midpoint of the diameter. Therefore, we can say that the distance from this midpoint to one of the opposite vertices of the square is the radius of the semicircle.
Now by looking at the construction above, we notice that the radius is also half of the diameter (also by definition). We see that d = s + 2a so it follows :
d = s +2a
r = (s + 2a)
r = s + a.
Furthermore, we know that the triangle formed by the radius, a side of the square, and half of the other side of the square (1/2s) is a right triangle. Since we have a right triangle, we can use the Pythagorean Theorem to solve for r.
By definition, the radius of a circle is a segment from the center of the circle to any point of the circle. So all radii are equal. As we observe above we have both equations equal to r, so we can set them equal and solve for the desired
Now divide both sides by a
Now solve for the desired ratio
If we multiply by the conjugate
We will get the golden ratio
Click here if you would like to explore with the GSP file of the Inscribed square in a half circle.