Answer:
13. x = 12
14. y = 4√3, z = 8√3
Explanation:
For this problem you don't need to know anything about geometric mean. You only need the Pythagorean theorem, and the knowledge that all of the right triangles are similar to each other.
The only value you can find initially is y. By the Pythagorean theorem, ...
4² + y² = 8²
y² = 64 -16 = 48
y = √48 = 4√3
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Now, we know the ratios of side lengths in these right triangles are ...
short leg : long leg : hypotenuse = 4 : 4√3 : 8 = 1 : √3 : 2
So ...
x = y√3 = (4√3)(√3) = 12
z = 2y = 8√3
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13. x = 12
14. y = 4√3, z = 8√3
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About geometric mean
You know the arithmetic mean is the sum of n numbers, divided by n.
The geometric mean is defined as the n-th root of the product of n numbers. When there are two numbers, it is the square root of their product.
The harmonic mean of n numbers is n divided by the sum of the reciprocals of the numbers.
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In this right triangle problem, the similarity geometry gives rise to several geometric mean relationships.
For example, y is the long side of ∆LMN and the short side of ∆LNO. Then we can write the ratio ...
long side : short side = y : 4 = x : y
Written as fractions, this becomes ...
y/4 = x/y
which resolves to ...
y = √(4x) . . . . . . y is the geometric mean of the lengths 4 and x (the parts of the longest hypotenuse)
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Similarly, z is both a long side and a hypotenuse, so we can write ratios involving those side lengths:
long side / hypotenuse = z/(x+4) = x/z
z = √(x(x +4)) . . . . . . z is the geometric mean of x and x+4
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And, likewise, we can look at the ratio ...
hypotenuse / short side = 8/4 = (x+4)/8
8 = √(4(x+4)) . . . . . . . 8 is the geometric mean of 4 and x+4
This relationship would allow us to solve for x in the absence of any other information:
64 = 4(x+4)
x = -4 +64/4 = 12
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In summary, the three geometric mean relationships in this right triangle geometry are ...
- the altitude to the hypotenuse is the geometric mean of the parts of the hypotenuse
- the short side of the large triangle is the geometric mean of the short hypotenuse segment and the whole hypotenuse
- the long side of the large triangle is the geometric mean of the long hypotenuse segment and the whole hypotenuse
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Personal note
Thank you for asking this question. In developing the geometric mean relationships in this triangle, I learned something new.