Question

A square with side length $l$ is inscribed in a semi-circle such that one side of the square is on the diameter of the semicircle. The perimeter of the semicircle is given by:

• A. $l\pi \sqrt{5}$
• B. $\frac{\pi \sqrt{5}l}{2}$
• C. $\sqrt{5}l(\frac{\pi }{2}+1)$
• D. $\frac{\sqrt{5}l}{2}(\pi +1)$

Answer 1

The correct option is C $\sqrt{5}l(\frac{\pi }{2}+1)$

Let M be the centre of circle of which semi circle is a part.

$In△AMP\&△MPQ:AM=MB=r.andPA=BQ=l.\angle APM=\angle BQM={90}^0.\therefore {△}^{\prime }sarecongruentbySAS.\Rightarrow MQ=MP=\frac{1}{2}PQ=\frac{l}{2}.In△BMQ:\Rightarrow r^2=l^2+(\frac{l}{2}{)}^2.r^2=l^2+\frac{l^2}{4}.r^2=\frac{{5l}^2}{4}.\Rightarrow r=\frac{\sqrt{5l}}{2}.Perimeterofsemicircle=\frac{1}{2}(2\pi r)+2r.=\pi r+2r.=\pi \frac{\sqrt{5l}}{2}+\sqrt{5l}.=\sqrt{5l}(\frac{\pi }{2}+1).\therefore C)Answer.$