Question

Find the information given in the figure. Show that ; $PM=PN=\sqrt{3}\times a$

Answer 1

From the fig

$QR=a\Rightarrow SQ=SR=\frac{a}{2}\therefore SN=SM=a+\frac{a}{2}=\frac{3a}{2}$

In equilateral $△PQR,PS$ is its altitude

$\therefore PS=\frac{\sqrt{3}}{2}a$

In right angled $△PSN$

$\therefore PS=\frac{\sqrt{3}}{2}a,SN=\frac{3}{2}a{PN}^2={PS}^2+{SN}^{2}PN=\sqrt{{\left(\frac{3a}{2}\right)}^2+{\left(dcfrac\sqrt{3}a2\right)}^2}=\sqrt{3}a$

Similarly in right $△PSM$

$PS=\frac{\sqrt{3}}{2}a,SM=\frac{3}{2}aPM=\sqrt{{PS}^2+{SM}^2}=\sqrt{3}a\therefore PM=PN=\sqrt{3}a$