In this assessment, you will be given the length of two segments, a and b. Your task will be to construct a line segment, x, whose length is the geometric mean between segments a and b.
Written instructions for constructing a segment whose length is the geometric mean between the lengths of the given segments
Constructing: The Geometric Mean
You have learned how to construct a geometric mean with a compass and a straight edge- now we are going to put that knowledge into practice. For this example, we will utilize those techniques to construct a segment with a length of the square root of 6, when we are given a segment labeled PQ whose length is 6.
Recall that in a geometric mean problem, the means of the proportion must be the same. If we cross-multiply, we get x-squared equals a times b and x equals the square root of a b.
[a over x = x over b
x squared = a b
x = square root of a b]
We know that we are looking for a segment that has the square root of 6 as its length. We can substitute the 6 into the last equation to get x equals the square root of 6. Squaring both sides gives us x-squared equals 6. Putting this information back into proportion form gives us 6 over x equals x over 1. Notice that we are not using 2 times 3 because the length of segment PQ is 6.
[x = square root of 6, x squared = 6, 1 over x = x over 6]
To make the construction we need to add the lengths of the extremes to have a single segment that is 7 units long and label it P R. Next, we will construct the perpendicular bisector of segment P R and use the point of intersection as the middle of our circle. Now, place the point of the compass on the intersection of our perpendicular and the segment P R. Extend the compass to either points P or R and draw the circle.
[Line P R. Point Q is on the line, and is 1 away from R. A circle is drawn intersecting points P and R.]
Lastly, we create a perpendicular at point Q and label the intersection with the circle S. The length of this perpendicular is the square root of 6. If we overlay the triangle P Q S, it is easy to see that the segment Q S is the geometric mean.