Question

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

Explanation:

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

In a right - triangle, if an altitude is drawn to the hypotenuse, then the length of each leg of the right - triangle is the geometric mean of the length of the hypotenuse and the segment of the hypotenuse adjacent to that leg.
For right - triangle \(QST\) with altitude \(TR\), we have the formula \(TS^{2}=SR\times SQ\).
Given \(SR = 12\) and \(SQ=6 + 12=18\).

Step2: Substitute the values into the formula

Substitute \(SR = 12\) and \(SQ = 18\) into \(TS^{2}=SR\times SQ\). Then \(TS^{2}=12\times18\).
Calculate \(12\times18=216\).

Step3: Find the value of \(TS\)

Take the square root of both sides: \(TS=\sqrt{216}\).
Simplify \(\sqrt{216}=\sqrt{36\times6}=6\sqrt{6}\).

Answer:

\(6\sqrt{6}\text{ units}\)