Given right triangle A BC with altitude B D drawn to hypotenuse AC. If AB = 6 and
AD = 4, what is the length of AC?
We can use the Pythagorean theorem to solve this problem.
Let AC = x and BC = y.
By Pythagorean theorem, we have:
AB^2 + BC^2 = AC^2
6^2 + y^2 = x^2
We also know that the length of the altitude BD is 4, which creates two similar right triangles: ADB and BDC.
Using the similarity of the triangles, we can set up the proportion:
AD/AB = BD/BC
4/6 = BD/y
2/3 = BD/y
Since BD = 4, we can solve for y using the proportion:
2/3 = 4/y
2y = 12
y = 6
Substituting this value for y back into the Pythagorean theorem equation:
6^2 + 6^2 = x^2
36 + 36 = x^2
72 = x^2
Taking the square root of both sides, we find:
x = √72
x = √(36*2)
x = 6√2
Therefore, the length of AC is 6√2.