Question Number 223935 by fantastic last updated on 10/Aug/25
Commented by fantastic last updated on 10/Aug/25
$${PERIMETER} \\ $$
Answered by dionigi last updated on 11/Aug/25
$${P}\:=\:\overset{\frown} {{AB}}\:+\:\overset{\frown} {{DE}}\:+{AD}+{EB} \\ $$$${P}\:=\:\pi{r}\:+\:\frac{\pi{r}}{\mathrm{2}}\:+\:{d}\:+\:\left({r}−{d}\right)\:=\:\frac{\mathrm{3}\pi{r}}{\mathrm{2}}\:+\:{r} \\ $$
Answered by mehdee7396 last updated on 12/Aug/25
$${DE}={AC}={r}\:\:\Rightarrow{P}={d}+{r}−{d}+\frac{\mathrm{3}}{\mathrm{2}}\pi{r}={r}+\frac{\mathrm{3}\pi}{\mathrm{2}}{r} \\ $$
Answered by MirHasibulHossain last updated on 12/Aug/25
![arc(AB)=(1/2)×2π×CB=πr [∵ AC=CB=r] CB=AC=r ⇒AC+CE=r+d ⇒AE=r+d ⇒AE−AD=r+d−d ⇒DE=r arc(DE)=(1/2)×2π×((DE)/2)=π×(r/2)=(1/2)πr EB=CB−CE=r−d Perimeter=AD+arc(DE)+EB+arc(AB) =d+(1/2)πr+r−d+πr =(3/2)πr+r =r((3/2)π+1) [ANSWER]](https://www.tinkutara.com/question/Q223994.png)
$$\mathrm{arc}\left(\mathrm{AB}\right)=\frac{\mathrm{1}}{\mathrm{2}}×\mathrm{2}\pi×\mathrm{CB}=\pi\mathrm{r}\:\:\:\:\:\left[\because\:\mathrm{AC}=\mathrm{CB}=\mathrm{r}\right] \\ $$$$\mathrm{CB}=\mathrm{AC}=\mathrm{r} \\ $$$$\Rightarrow\mathrm{AC}+\mathrm{CE}=\mathrm{r}+\mathrm{d} \\ $$$$\Rightarrow\mathrm{AE}=\mathrm{r}+\mathrm{d} \\ $$$$\Rightarrow\mathrm{AE}−\mathrm{AD}=\mathrm{r}+\mathrm{d}−\mathrm{d} \\ $$$$\Rightarrow\mathrm{DE}=\mathrm{r} \\ $$$$\mathrm{arc}\left(\mathrm{DE}\right)=\frac{\mathrm{1}}{\mathrm{2}}×\mathrm{2}\pi×\frac{\mathrm{DE}}{\mathrm{2}}=\pi×\frac{\mathrm{r}}{\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{2}}\pi\mathrm{r} \\ $$$$\mathrm{EB}=\mathrm{CB}−\mathrm{CE}=\mathrm{r}−\mathrm{d} \\ $$$$\mathrm{Perimeter}=\mathrm{AD}+\mathrm{arc}\left(\mathrm{DE}\right)+\mathrm{EB}+\mathrm{arc}\left(\mathrm{AB}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{d}+\frac{\mathrm{1}}{\mathrm{2}}\pi\mathrm{r}+\mathrm{r}−\mathrm{d}+\pi\mathrm{r} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{3}}{\mathrm{2}}\pi\mathrm{r}+\mathrm{r} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{r}\left(\frac{\mathrm{3}}{\mathrm{2}}\pi+\mathrm{1}\right)\:\:\:\:\:\:\:\:\:\:\left[\mathrm{ANSWER}\right] \\ $$