Question

The parallelogram PQRS is formed by joining together four equilateral triangles of side 1 unit, as shown in the figure.

What is the length of the diagonal SQ?

• A. $\sqrt{7}units$
• B. $\sqrt{8}units$
• C. $\sqrt{6}units$
• D. $\sqrt{5}units$

Answer 1

The correct option is A

$\sqrt{7}units$

$RT=\frac{1}{2}units,SR=1+1=2units$
$ST=(2+\frac{1}{2})units=\frac{5}{2}units$
$QT=\sqrt{Q{R}^2-R{T}^2}=\sqrt{1^2-{\left(\frac{1}{2}\right)}^2}=\frac{\sqrt{3}}{2}units$
Now in $\Delta SQT$, SQ is the hypotenuse.
So, $SQ=\sqrt{S{T}^2+Q{T}^2}=\sqrt{{\left(\frac{5}{2}\right)}^2+{\left(\frac{\sqrt{3}}{2}\right)}^2}$
$=\sqrt{7}units$
Therefore, the length of the diagonal SQ is $=\sqrt{7}units$