Step-by-step explanation:
In order to find the length of side TS, first we need to find the length of side TR which is perpendicular to side QS.
By Geometric mean property:
[tex] {TR} = \sqrt{6 \times 12} \\ \therefore \: {TR} = \sqrt{6 \times 6 \times 2} \\ \therefore \: {TR} = 6\sqrt{2} \\ \\ In \: right \: \triangle \: TRS: \\ TS^2 = TR^2 + RS^2 \\ \therefore \: (3x)^{2} = {(6 \sqrt{2}) }^{2} + {(12)}^{2} \\ \therefore \: 9x^{2} = 72 + 144 \\ \therefore \: 9x^{2} =2 1 6 \\ \therefore \: x^{2} = \frac{2 1 6}{9} \\ \therefore \: x^{2} =24 \\ \therefore \: x = \sqrt{24} \\ \therefore \: x = 4.89897949 \\ \therefore \: x \approx \: 4.9 \\ \therefore \: 3x = 3 \times 4.9 \\ \therefore \: 3x =14.7 \\ \huge \red{ \boxed{\therefore \: TS = 14.7 \: units}}[/tex]
