Question

In triangle △ABC, ∠ABC=90°,

Answer 1

Answer: As a fraction, HC is 21/4 units long

In decimal form, HC is 5.25 units long.

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Work Shown:

Let x = HC and y = BH

AH = 12-x since AH = 12-HC and x = HC

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See the attached drawing below. The altitude BH forms three similar triangles which makes the proportion below possible

HC/BH = BH/AH

x/y = y/(12-x) ... substitution

x(12-x) = y^2 ... cross multiply

y^2 = -x^2+12x

note how I isolated y^2 instead of y. We will use this equation later.

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Focus on triangle AHB. Use the pythagorean theorem

(BH)^2 + (AH)^2 = (AB)^2

(y)^2 + (12-x)^2 = (9)^2

y^2 + 144-24x+x^2 = 81

-x^2+12x + 144-24x+x^2 = 81 ... replace y^2 with -x^2+12x

-x^2+12x + 144-24x+x^2-81 = 0

-12x + 63 = 0

-12x = -63

x = -63/(-12)

x = 5.25 <<-- answer in decimal form

x = 5 + 0.25

x = 5 + 1/4

x = 20/4 + 1/4

x = 21/4 <<-- answer as a fraction