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Question Number 211340 by Tawa11 last updated on 06/Sep/24

Commented by Frix last updated on 06/Sep/24

$$\mathrm{Question}211294$$

Answered by mr W last updated on 06/Sep/24

$$s=\frac{a+b+c+d}{2}=\frac{1+2+3+4}{2}=5$$$$\left[ABCD\right]=\sqrt{(s-a)(s-b)(s-c)(s-d)-abcd{\cos }^2\left(\frac{x+y}{2}\right)}$$$$=\sqrt{4\times 3\times 2\times 1-1\times 2\times 3\times 4{\cos }^{2}60°}$$$$=\sqrt{24\times (1-\frac{1}{4})}=3\sqrt{2}✓$$

Commented by Tawa11 last updated on 06/Sep/24

$$\mathrm{Wow}.$$$$\mathrm{Thank}\mathrm{you}\mathrm{sir}.$$$$\mathrm{I}\mathrm{really}\mathrm{appreciate}.$$

Commented by mr W last updated on 06/Sep/24

Commented by mr W last updated on 06/Sep/24

Commented by mr W last updated on 06/Sep/24

$$seealsoQ30233$$

Answered by mahdipoor last updated on 06/Sep/24

$${BD}^2=a^2+d^2-2ad.cosx=b^2+c^2-2bc.cos(120-x)$$$$\Rightarrow 17-8cosx=13-12cos(120-x)$$$$\Rightarrow 4=8cosx-12(-\frac{1}{2}cosx+\frac{\sqrt{3}}{2}sinx)$$$$\Rightarrow 2=7c-3\sqrt{3}seqi$$$$S=\frac{ad}{2}sin\left(x\right)+\frac{bc}{2}sin(120-x)=$$$$2sin\left(x\right)+3(\frac{\sqrt{3}}{2}cos\left(x\right)-\frac{1}{2}sin\left(x\right))=$$$$\Rightarrow 2S=s+3\sqrt{3}ceqj$$$$i,j\Rightarrow \{\begin{array}{l}c=\frac{1+3\sqrt{3}S}{17}\\ s=\frac{7S-3\sqrt{3}}{17}\end{array}\Rightarrow$$$$1=c^2+s^2=\frac{28+76S^2-36\sqrt{3}S}{{17}^2}$$$$\Rightarrow 76S^2-36\sqrt{3}S-261=0\Rightarrow S=\frac{9\sqrt{3}+8\sqrt{129}}{38}!?$$

Commented by Frix last updated on 06/Sep/24

$$S=2\sin +3(\frac{\sqrt{3}}{2}\cos x+\frac{1}{2}\sin x)$$$$2S=7s+3\sqrt{3}c$$$$\Rightarrow$$$$\{\begin{array}{l}c=\frac{7+3\sqrt{3}S}{38}\\ s=\frac{7S+3\sqrt{3}}{38}\end{array}$$$$\Rightarrow S=3\sqrt{2}$$

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