Question

solve for x and y.

x = 2\sqrt{3} (simplify your answer. type an exact answer, using radicals as needed.)
y = \square (simplify your answer. type an exact answer, using radicals as needed.)

Explanation:

Step1: Use the geometric mean theorem (altitude-on-hypotenuse theorem)

For a right - triangle, if an altitude \(x\) is drawn to the hypotenuse, then \(x=\sqrt{6\times2}\). But we already know \(x = 2\sqrt{3}\). For the segment \(y\) of the hypotenuse, we use the geometric mean theorem: \(y=\frac{x^{2}}{2}\).
Substitute \(x = 2\sqrt{3}\) into the formula: \(y=\frac{(2\sqrt{3})^{2}}{2}\).

Step2: Simplify the expression

First, expand \((2\sqrt{3})^{2}=4\times3 = 12\). Then, \(\frac{12}{2}=6\). Another way is to use the geometric mean theorem \(y^{2}=x^{2}+(2\sqrt{3})^{2}\) (by Pythagorean theorem in the right - triangle with sides \(x\) and \(2\sqrt{3}\) and hypotenuse \(y\)). But using the proportion from similar triangles: In right - triangle, if the hypotenuse is \(6 + 2=8\), and by the geometric mean \(y^{2}=6\times8\).

$$y=\sqrt{6\times8}=\sqrt{48}=4\sqrt{3}$$

Answer:

\(y = 4\sqrt{3}\)