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Question Number 205406 by mr W last updated on 20/Mar/24

Commented by lepuissantcedricjunior last updated on 21/Mar/24

$${\mathit{d}}^{\prime }{\mathit{a}\mathit{p}\mathit{r}\mathit{e}}^{\prime }\mathit{s}\mathit{p}\mathit{y}\mathit{t}\mathit{h}\mathit{a}\mathit{g}\mathit{o}\mathit{r}\mathit{e}$$$${\mathit{d}}^2=7^2+{\mathit{k}}^2\left(1\right)$$$${\mathit{d}}^{\prime }\mathit{a}\mathit{p}\mathit{r}\mathit{e}\mathit{s}\mathit{a}\mathit{l}\mathit{k}\mathit{a}\mathit{s}\mathit{h}\mathit{i}\mathit{l}$$$${15}^2+{15}^2={\mathit{k}}^2+2\times {15}^2\mathit{c}\mathit{o}\mathit{s}(\mathit{\pi }-\mathit{a})$$$$2.{15}^2={\mathit{k}}^2-2.{15}^2\mathit{c}\mathit{o}\mathit{s}\mathit{a}$$$${\mathit{k}}^2=2.{15}^2(1+{\mathit{c}\mathit{o}\mathit{s}}^2\mathit{a})\left(2\right)$$$$\frac{\mathit{k}}{\mathit{s}\mathit{i}\mathit{n}(\mathit{\pi }-\mathit{a})}=\frac{15}{\mathit{s}\mathit{i}\mathit{n}\mathit{a}}=\frac{{15}^2\mathit{k}}{2\mathit{A}}=2r=d$$$$\frac{\mathit{k}}{\mathit{c}\mathit{o}\mathit{s}\mathit{a}}=\frac{15}{\mathit{s}\mathit{i}\mathit{n}\mathit{a}}=\frac{{15}^2.\mathit{k}}{2\mathit{A}}=\mathit{d}$$$$\mathit{c}\mathit{o}\mathit{s}\left(\mathit{a}\right)=\frac{\mathit{k}}{\mathit{d}}\left(2\right)\mathit{r}\mathit{e}\mathit{m}\mathit{p}\mathit{l}\mathit{a}\mathit{c}\mathit{o}\mathit{n}\mathit{s}\mathit{d}\mathit{a}\mathit{n}\mathit{s}\left(2\right)$$$${\mathit{k}}^2=2.{15}^2(1+\frac{{\mathit{k}}^2}{{\mathit{d}}^2})=\frac{{\left(15\mathit{d}\sqrt{2}\right)}^2+{\left(15\sqrt{2}\mathit{k}\right)}^2}{{\mathit{d}}^2}$$$${\mathit{k}}^2{\mathit{d}}^2-{\left(15\sqrt{2}\right)}^2{\mathit{k}}^2=2.{15}^2{\mathit{d}}^2$$$${\mathit{k}}^2=\frac{2.{15}^2{\mathit{d}}^2}{{\mathit{d}}^2-2.{15}^2}\left(3\right)\mathit{r}\mathit{e}\mathit{m}\mathit{p}\mathit{l}\mathit{a}\mathit{c}\mathit{o}\mathit{n}\mathit{s}\mathit{d}\mathit{a}\mathit{n}\mathit{s}\left(1\right)$$$${\mathit{d}}^2=7^2+\frac{2.{15}^2{\mathit{d}}^2}{{\mathit{d}}^2-2.{15}^2}$$$$=>{\mathit{d}}^4-{\mathit{d}}^2(7^2+2.{15}^2)+2.{(7.15)}^2=0$$$$\iff {\mathbf{d}}^4-{\mathbf{d}}^2(49+450)+2{\left(105\right)}^2=0$$$$\mathbf{posons}{\mathbf{d}}^2=\mathbf{X}=>{\mathit{d}}^4={\mathit{X}}^2$$$${\mathbf{X}}^2-499\mathbf{X}+22050=0$$$$△=249001-88200=160801={401}^2$$$$\mathbf{X}=\frac{499-401}{2}=49$$$$\mathrm{X}=\frac{499+401}{2}=450$$$$or\mathbf{X}={\mathit{d}}^2=>\mathit{d}=\sqrt{49}=7\mathit{i}\mathit{m}\mathit{p}\mathit{o}\mathit{s}\mathit{s}\mathit{i}\mathit{b}\mathit{l}\mathit{e}$$$$\mathit{d}=\sqrt{450}=15\sqrt{2}$$$$\mathit{d}\mathit{o}\mathit{u}$$$$\overline{)\begin{array}{c}\mathit{d}\ne 7\\ \mathit{d}=15\sqrt{2}\sqrt{}\end{array}}$$$$=============================$$$$...............lepuissantD\mathit{r}.........$$$$$$$$$$$$$$

Commented by mr W last updated on 21/Mar/24

$$pleasepostyouranswertoaquestion$$$$as‘‘{answer}^{″},notas‘‘{comment}^{″}!thanks!$$$$btw,youranswer15\sqrt{2}iswrong.$$

Answered by A5T last updated on 20/Mar/24

$$\sqrt{{15}^2+{15}^2-2\times {15}^{2}cos\theta }=\sqrt{7^2+d^2+14dcos\theta }$$$$\Rightarrow 401-450cos\theta =d^2+14dcos\theta \Rightarrow cos\theta =\frac{d^2-401}{-450-14d}$$$$r^2=7^2+r^2+2\times 7rcos\theta \Rightarrow dcos\theta =-7\Rightarrow cos\theta =\frac{-7}{d}$$$$\Rightarrow \frac{d^2-401}{-450-14d}=\frac{-7}{d}\Rightarrow d=25$$

Commented by A5T last updated on 20/Mar/24

Commented by A5T last updated on 20/Mar/24

$$Generalformula,takea=7,b=15,c=15,d=2R$$

Commented by mr W last updated on 20/Mar/24

$$thanks!$$

Answered by mr W last updated on 20/Mar/24

Commented by mr W last updated on 20/Mar/24

$$R^2-{\left(\frac{7}{2}\right)}^2={15}^2-{(R-\frac{7}{2})}^2$$$$2R^2-7R-{15}^2=0$$$$R=\frac{7+\sqrt{49+8\times {15}^2}}{4}=12.5$$$$\Rightarrow d=2R=25✓$$

Answered by cortano12 last updated on 20/Mar/24

$$\Rightarrow 7^2+{15}^2+{15}^2+\frac{7.15.15}{\mathrm{R}}={4\mathrm{R}}^2$$$$\Rightarrow 499+\frac{1575}{\mathrm{R}}={4\mathrm{R}}^2$$$$\Rightarrow {4\mathrm{R}}^3-499\mathrm{R}-1575=0$$$$\Rightarrow (2\mathrm{R}-25)(\mathrm{R}+9)(2\mathrm{R}+7)=0$$$$\Rightarrow \mathrm{d}=2\mathrm{R}=25$$

Commented by mr W last updated on 20/Mar/24

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