Carcass wrote:

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 \(RS = \frac{1}{4\sqrt3} *\frac{\sqrt3}{\sqrt3} = \frac{\sqrt3}{12}\)

Attachments

FIG 1.jpg [ 17.51 KiB | Viewed 1885 times ]

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