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 is drawn to the hypotenuse, then \(TR^{2}=QR\times RS\). Given \(QR = 6\) and \(RS=12\), so \(TR^{2}=6\times12 = 72\), then \(TR=\sqrt{72}=6\sqrt{2}\).

Step2: Use Pythagorean theorem in \(\triangle TRS\)

In right - triangle \(\triangle TRS\), by \(a^{2}+b^{2}=c^{2}\) (\(a = TR\), \(b = RS\), \(c = TS\)). We know \(TR = 6\sqrt{2}\), \(RS = 12\). Then \(TS=\sqrt{(6\sqrt{2})^{2}+12^{2}}=\sqrt{72 + 144}=\sqrt{216}=6\sqrt{6}\).

Answer:

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