I want the value of HC in the following example:
The pictures shows a rectangle triangle (a = 10, b = X, c = 8), with the special point H in the hypothenuse, marked by height.I want to find the distance value between point H to point C.
It's safe to jump to this point of the exercise:
The X side is the result from the Pythagoras theorem where: 10² = 8² + X² = > X = 6
Got height H from: hypotenuse * height = opposite *adjacent => 10 * H = 6 * 8 = > H = 4.8
Now I can follow two different ways:
Use pythagoras theorem to find the value of HC, 8² = 4.8² + HC² => HC = 6.4
Or: Use hypotenuse projection as in 6 / AH = 8 / HC. and AH + HC = 10
AH = 10 - HC = > 6 / (HC-10) = 8 / HC => HC = 5.7
Hypotenuse projection gives me HC = 5.7, Pythagoras Theorem gives me HC = 6.4.
Where did I messed up? Which one is the correct answer?
It should be $\frac{6}{AH}=\frac{8}{BH} \implies AH=0.75\cdot BH=3.6$ which matches Pythagoras. This is because $\angle A \cong \angle HBC$. So you've made a mistake using hypotenuse projection.