Question

triangle PQR is an equilateral triangle with each side of length 2p.If PS is perpendicular to QR then find the value of PS.

Answer 1

$$\begin{gathered} \mathrm{Area}\mathrm{of}\mathrm{equilateral}\mathrm{triangle}\mathrm{ABC}\hspace{0.17em}=\frac{\sqrt{3}}{4}{\left(\mathrm{side}\right)}^2 \\ =\frac{\sqrt{3}}{4}\times {\left(2\mathrm{p}\right)}^2=\sqrt{3}{\mathrm{p}}^2 \\ \mathrm{Also}\mathrm{PS}\hspace{0.17em}\mathrm{is}\mathrm{perpendicular}\mathrm{and}\mathrm{QR}\mathrm{is}\mathrm{base}\mathrm{so} \\ \mathrm{Area}\mathrm{of}\mathrm{equilateral}\mathrm{triangle}\mathrm{ABC}=\frac{1}{2}\times \mathrm{Base}\times \mathrm{Height} \\ =\frac{1}{2}\times 2\mathrm{p}\times \mathrm{QR}=\mathrm{p}\times \mathrm{QR} \\ \mathrm{So},\mathrm{p}\times \mathrm{QR}=\sqrt{3}{\mathrm{p}}^2 \\ \mathrm{QR}=\sqrt{3}\mathrm{p}\mathrm{unit} \end{gathered}$$