Exploring and Defining Geometric Mean (Altitude)
What is the sum of all interior angles in any triangle?
What is the sum of the measures of the red and blue angles? How do you know this to be true?
The segment that was drawn as you dragged the slider is called an altitude. This altitude was drawn to the hypotenuse. How many right triangles did this altitude split the original right triangle into? (Please name the new smaller right triangles)
What does the the special movement of the red and blue angles imply about these 2 smaller right triangles? What previously learned theorem justifies your answer?
Does your response for the previous question also hold true for the relationship between the ORIGINAL BIG RIGHT TRIANGLE () and either one of the smaller right triangles? If so, how/why do you know this?
Compare to and finish the proportion.
The geometric mean of two numbers is the number x that satisfies such that . Look back at your observations. What two values is CD the geometric mean of?
Look at the two values you answered in the previous question. Using the animation, describe how your two values relate to the hypotenuse of the original triangle.
Use the observations you made during this exploration to finish writing the theorem below.
In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the _______ is the ________ ________ of the lengths of the two segments of the _________.
Type the value you found for x below.
Type the value you found for VX below.