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If \(PQ = 1\), what is the length of \(RS\)
A. \(\frac{1}{12}\)
B. \(\frac{\sqrt{3}}{12}\)
C. \(\frac{1}{6}\)
D. \(\frac{2}{3 \sqrt{3}}\)
E. \(\frac{2}{\sqrt{12}}\)
Here PQ = 1
Plz see the diagram attached.
Now △ PQT,
∠QPT = 30° , ∠QTP = 90° and ∠PQT = 60°so it is a 30° - 60° -90° and we know the sides are distributed in the ratio \(1 : \sqrt3 : 2\)
Now PQ = 1,
so
\(QT = \frac{1}{2} and PT = \frac{2}{sqrt3}\)Now let us consider △ QTS
∠TQS = 30° , ∠QST = 60° and ∠QTS = 90°so it is a 30° - 60° -90° and we know the sides are distributed in the ratio \(1 : \sqrt3 : 2\)
Now \(QT = \frac{1}{2}\),
so
\(QS = \frac{1}{\sqrt3} and TS = \frac{1}{2\sqrt3}\)Now let us consider △ TRS
∠RST = 60° , ∠RTS = 30° and ∠TRS = 90°so it is a 30° - 60° -90° and we know the sides are distributed in the ratio \(1 : \sqrt3 : 2\)
Now \(TS = \frac{1}{2\sqrt3}\),
so
\(RT = \frac{1}{4} and RS = \frac{1}{4\sqrt3}\).
But \(RS = \frac{1}{4\sqrt3}\) can also be written as \(RS = \frac{1}{4\sqrt3} *\frac{\sqrt3}{\sqrt3} = \frac{\sqrt3}{12}\)
Is it not PT = Sqrt(3)/2? If it is 2/sqrt(3), what am I doing wrong to get sqrt(3)/2?