Question

for each of the following, solve for the unknowns.
a. b.
a. ( x=square ) and ( y=6 sqrt{3} )
(type exact answers, using radicals as needed.)

Explanation:

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

In a right - triangle, if an altitude \(y\) is drawn to the hypotenuse \(x\) from the right - angle vertex, then \(y^{2}\) is the geometric mean of the segments of the hypotenuse. Also, the length of a leg of the right - triangle is the geometric mean of the hypotenuse and the adjacent segment.
Let the hypotenuse be \(x\). The two legs of the large right - triangle are \(a = 12\sqrt{3}\) and \(b = 12\).
We know the formula \(a^{2}=x\times\) (segment adjacent to \(a\)), \(b^{2}=x\times\) (segment adjacent to \(b\)), and \(y^{2}=\) (segment adjacent to \(a\))\(\times\) (segment adjacent to \(b\)). Another formula is \(x=\frac{a^{2}+b^{2}}{x}\) (by Pythagoras \(a^{2}+b^{2}=x^{2}\)).
First, calculate \(a^{2}=(12\sqrt{3})^{2}=144\times3 = 432\) and \(b^{2}=12^{2}=144\).
By Pythagoras' theorem \(x^{2}=a^{2}+b^{2}\).

Step2: Solve for \(x\)

Substitute \(a^{2}=432\) and \(b^{2}=144\) into \(x^{2}=a^{2}+b^{2}\).
\(x^{2}=432 + 144=576\).
Take the square root of both sides: \(x=\sqrt{576}=24\).

Answer:

\(x = 24\)