Question Number 3836 by Rasheed Soomro last updated on 22/Dec/15
$${A}\:{square},{whose}\:{area}\:{is}\:{s}^{\mathrm{2}} ,{contains}\: \\ $$$${a}\:{semicircle}\:{of}\:{possible}\:{largest}\:{area}. \\ $$$${Determine}\:{radius}\:{of}\:{the}\:{semicircle}. \\ $$
Commented by Yozzii last updated on 24/Dec/15
$${r}=\frac{{s}}{\mathrm{2}}.\:{I}\:{think}\:{that}\:{in}\:{order}\:{to}\: \\ $$$${obtain}\:{a}\:{semi}−{circle}\:{of}\:{largest}\:{area} \\ $$$${in}\:{a}\:{square},\:{it}\:{is}\:{necessary}\:{that}\:{its} \\ $$$${straight}\:{edge}\:{is}\:{parallel}\:{to}\:{one}\:{of}\:{the} \\ $$$${sides}\:{of}\:{the}\:{square}.\:{If}\:{any}\:{larger}\:{side}\:{length} \\ $$$${is}\:{sought}\:{after},\:{part}\:{of}\:{the}\:{semicircle} \\ $$$${lies}\:{outside}\:{the}\:{square}.\:{In}\:{other}\:{words}, \\ $$$${if}\:{the}\:{arm}\:{representing}\:{the}\:{side}\:{edge}\:{of}\:{the}\:{semicircle}\:{is} \\ $$$${free}\:{to}\:{rotate}\:\theta°\:{within}\:{the}\:{region}\:{of}\:{the} \\ $$$${square}\:{about}\:{one}\:{of}\:{the}\:{corners}\:{of}\:{the} \\ $$$${square}\:\left(\theta={angle}\:{between}\:{arm}\:{and}\:{side}\:{of}\:{square}\right),\:{only}\:{when}\:\theta=\mathrm{0}°\:{can}\:{we}\: \\ $$$${completely}\:{contain}\:{a}\:{semi}−{circle}\:{in}\: \\ $$$${the}\:{fixed}\:{square}.\:{So}\:{s}=\mathrm{2}{r}\Rightarrow{r}=\frac{{s}}{\mathrm{2}}. \\ $$
Commented by RasheedSindhi last updated on 24/Dec/15
$${Straight}\:{edge}\:{of}\:{semicircle}\:{is}\:{parallel}\:{to}\:{one} \\ $$$${of}\:{the}\:{diagonals}\:{of}\:{the}\:{square}\:{and} \\ $$$${it}\:{touches}\:{all}\:{the}\:{sides}\:\:{of}\:{square}. \\ $$
Commented by RasheedSindhi last updated on 24/Dec/15
$${Drawing}\:{a}\:{semicircle}\:\left(\:{by}\:{ruler}\right. \\ $$$$\left.{and}\:{compass}\right)\:{in}\:{a}\:{square}\:{that} \\ $$$${touches}\:{all}\:{the}\:{sides}\:{of}\:{the} \\ $$$${square}\:{is}\:{a}\:{familiar}\:{construction} \\ $$$${problem}. \\ $$
Commented by Yozzii last updated on 24/Dec/15
$${Hmm}.\:{Ok}.\: \\ $$
Commented by Rasheed Soomro last updated on 29/Dec/15
