Exploring the Geometric Mean with Similar Triangles

Geometric Mean Discovery Applet In this applet we explore the geometric mean as it relates to similar triangles, and the relationships between the sides in our construction.

How many triangles are on screen? Notice what happens when you move points D and A. What stays the same? How do the triangles change? (Hint: Consider similar triangles and side ratios) Can you find multiple triangles sets where C is a vertice? From the rest of the questions, assume the constructions has C aligned to be a vertice of a triangle. Come up with some sets of side length ratios that are equal to one another. How many ways can you come up with to write the length of PB by using these ratios? (Without using the actual length measurement) Were any of these geometric means? (Note: A geometric mean is the square root of a product of two or more numbers) Were there any that didn’t seem like geometric means? Can you describe the square of PA’s length using other side lengths? Challenge: Can you describe PA’s length in terms of just PC and PD? What does this length represent? (Hint: You may need to use two geometric means.) How would you make sure PD is the same length as PC? Double the length? Eight times the length of PC? If PC is length 1 and PD is length 8, what is the length of PB? What is the length of PA? (Hint: Consider your ratios, which sets of ratios or fractions do you think you should use?) What if PD is length 2, and PC is length 1, then what is the length of PB? What if PD is any constant multiple of PC? What do we think that this construction could be used for? What is significant about the length of PB in relation to lengths PD and PA? How could this be useful for doubling the volume of an object?