Question

In triangle △ABC, ∠ABC=90°, BH = altitude

Answer 1

Answer: The exact length of segment HC is sqrt(3) units

The approximate length is roughly 1.73205080756888 (round that however you need to)

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Work Shown:

Let x = length of HC

Since AH = 3*HC, this means AH = 3*x

Draw out the picture. This step is optional but helpful in my opinion. The drawing is attached below.

After adding in the altitude BH, we have three similar triangles. So we can form the proportion shown below to solve for x

HC/BH = BH/AH

HC/3 = 3/AH ... replace BH with 3

x/3 = 3/AH ... replace HC with x

x/3 = 3/(3x) ... replace AH with 3x

x/3 = 1/x ... reduce

x*x = 3*1 ... cross multiply

x^2 = 3

x = sqrt(3) ... which is shorthand for "square root"

HC = sqrt(3)

HC = 1.73205080756888 which is approximate