Answer:
2. 6√2
Explanation:
You want the measure of altitude GI that divides the hypotenuse of right triangle FGH into the parts FI=6 and IH=12.
Geometric mean
The altitude of the triangle is the geometric mean of the lengths of the hypotenuse segments:
GI = √(FI·IH)
GI = √(6·12) = 6√2
The length of GI is 6√2.
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Additional comment
This relation is based on the fact that all of the triangles are similar. The ratios of short to long sides are ...
FI/GI = GI/IH
GI² = FI·IH . . . . . . . cross multiply
GI = √(FI·IH) . . . . . take the square root
The two other geometric mean relations in this geometry are ...
FG = √(FI·FH
HG = √(HI·HF)
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