Question

what is the length of side ts?
2√6 units
6√6 units
24 units
8 units

Explanation:

Step1: Apply geometric mean theorem

In a right - triangle, if an altitude \(TR\) is drawn to the hypotenuse \(QS\) of \(\triangle QTS\) (\(\angle QTS = 90^{\circ}\)), then \(TR^{2}=QR\times RS\). Let's assume we use the relationship for the sides of the right - triangle. According to the geometric mean (altitude) theorem, for a right - triangle \(\triangle QTS\) with altitude \(TR\) to the hypotenuse \(QS=(6 + 12)=18\), and another relationship: \(TS^{2}=RS\times QS\) (by the geometric mean (leg) theorem: in a right - triangle, the square of a leg is equal to the product of the hypotenuse and the adjacent segment of the hypotenuse).

Step2: Substitute values

We know that \(RS = 12\) and \(QS=6 + 12=18\). Substitute these values into the formula \(TS^{2}=RS\times QS\). So \(TS^{2}=12\times18\).

Step3: Calculate \(TS^{2}\) and \(TS\)

First, \(12\times18=(2\times6)\times(2\times9)=2^{2}\times54\). Then \(TS^{2}=216\). Since \(TS=\sqrt{216}\), and \(216 = 36\times6\), so \(TS=\sqrt{36\times6}\). Using the property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}(a = 36,b = 6)\), we get \(TS = 6\sqrt{6}\) units.

Answer:

B. \(6\sqrt{6}\) units