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Question Number 164154 by HongKing last updated on 14/Jan/22

{ ((x^2 + xy + y^2 = 9)),((y^2 + yz + z^2 = 16)),((x^2 + xz + z^2 = 25)) :} Find: S = xy + yz + xz

$$\begin{cases}{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{xy}\:+\:\mathrm{y}^{\mathrm{2}} \:=\:\mathrm{9}}\\{\mathrm{y}^{\mathrm{2}} \:+\:\mathrm{yz}\:+\:\mathrm{z}^{\mathrm{2}} \:=\:\mathrm{16}}\\{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{xz}\:+\:\mathrm{z}^{\mathrm{2}} \:=\:\mathrm{25}}\end{cases} \\ $$$$\mathrm{Find}:\:\:\:\mathrm{S}\:=\:\mathrm{xy}\:+\:\mathrm{yz}\:+\:\mathrm{xz} \\ $$

Answered by behi834171 last updated on 14/Jan/22

p=((3+4+5)/2)=6 A=(√(6×1×2×3))=6 6=((√3)/4)(xy+yz+zx)⇒ xy+yz+zx=8(√3) .■

$${p}=\frac{\mathrm{3}+\mathrm{4}+\mathrm{5}}{\mathrm{2}}=\mathrm{6} \\ $$$${A}=\sqrt{\mathrm{6}×\mathrm{1}×\mathrm{2}×\mathrm{3}}=\mathrm{6} \\ $$$$\mathrm{6}=\frac{\sqrt{\mathrm{3}}}{\mathrm{4}}\left(\boldsymbol{{xy}}+\boldsymbol{{yz}}+\boldsymbol{{zx}}\right)\Rightarrow \\ $$$$\:\:\:\:\:\:\:\boldsymbol{{xy}}+\boldsymbol{{yz}}+\boldsymbol{{zx}}=\mathrm{8}\sqrt{\mathrm{3}}\:\:\:.\blacksquare \\ $$

Commented by mr W last updated on 15/Jan/22

very nice solution!

$${very}\:{nice}\:{solution}! \\ $$

Commented by HongKing last updated on 15/Jan/22

thank you so much dear Sir

$$\mathrm{thank}\:\mathrm{you}\:\mathrm{so}\:\mathrm{much}\:\mathrm{dear}\:\mathrm{Sir} \\ $$

Commented by Rasheed.Sindhi last updated on 15/Jan/22

Explain 3rd line please!

$$\mathcal{E}{xplain}\:\mathrm{3}{rd}\:{line}\:{please}! \\ $$

Commented by mr W last updated on 15/Jan/22

geometry meaning of the solution see Q164174

$${geometry}\:{meaning}\:{of}\:{the}\:{solution} \\ $$$${see}\:{Q}\mathrm{164174} \\ $$

Commented by Tawa11 last updated on 15/Jan/22

Nice sir

$$\mathrm{Nice}\:\mathrm{sir} \\ $$

Commented by Rasheed.Sindhi last updated on 15/Jan/22

ThanX mr W sir!

$$\mathcal{T}{han}\mathcal{X}\:{mr}\:{W}\:\boldsymbol{{sir}}! \\ $$

Commented by mr W last updated on 15/Jan/22

we get further x=((4(√6)×6+3(√2)(−3^2 +4^2 +5^2 ))/(6(√(3^2 +4^2 +5^2 +4(√3)×6))))=((16(√2)+4(√6))/( (√(50+24(√3))))) y=((4(√6)×6+3(√2)(3^2 −4^2 +5^2 ))/(6(√(3^2 +4^2 +5^2 +4(√3)×6))))=((9(√2)+4(√6))/( (√(50+24(√3))))) z=((4(√6)×6+3(√2)(3^2 +4^2 −5^2 ))/(6(√(3^2 +4^2 +5^2 +4(√3)×6))))=((4(√6))/( (√(50+24(√3)))))

$${we}\:{get}\:{further} \\ $$$${x}=\frac{\mathrm{4}\sqrt{\mathrm{6}}×\mathrm{6}+\mathrm{3}\sqrt{\mathrm{2}}\left(−\mathrm{3}^{\mathrm{2}} +\mathrm{4}^{\mathrm{2}} +\mathrm{5}^{\mathrm{2}} \right)}{\mathrm{6}\sqrt{\mathrm{3}^{\mathrm{2}} +\mathrm{4}^{\mathrm{2}} +\mathrm{5}^{\mathrm{2}} +\mathrm{4}\sqrt{\mathrm{3}}×\mathrm{6}}}=\frac{\mathrm{16}\sqrt{\mathrm{2}}+\mathrm{4}\sqrt{\mathrm{6}}}{\:\sqrt{\mathrm{50}+\mathrm{24}\sqrt{\mathrm{3}}}} \\ $$$${y}=\frac{\mathrm{4}\sqrt{\mathrm{6}}×\mathrm{6}+\mathrm{3}\sqrt{\mathrm{2}}\left(\mathrm{3}^{\mathrm{2}} −\mathrm{4}^{\mathrm{2}} +\mathrm{5}^{\mathrm{2}} \right)}{\mathrm{6}\sqrt{\mathrm{3}^{\mathrm{2}} +\mathrm{4}^{\mathrm{2}} +\mathrm{5}^{\mathrm{2}} +\mathrm{4}\sqrt{\mathrm{3}}×\mathrm{6}}}=\frac{\mathrm{9}\sqrt{\mathrm{2}}+\mathrm{4}\sqrt{\mathrm{6}}}{\:\sqrt{\mathrm{50}+\mathrm{24}\sqrt{\mathrm{3}}}} \\ $$$${z}=\frac{\mathrm{4}\sqrt{\mathrm{6}}×\mathrm{6}+\mathrm{3}\sqrt{\mathrm{2}}\left(\mathrm{3}^{\mathrm{2}} +\mathrm{4}^{\mathrm{2}} −\mathrm{5}^{\mathrm{2}} \right)}{\mathrm{6}\sqrt{\mathrm{3}^{\mathrm{2}} +\mathrm{4}^{\mathrm{2}} +\mathrm{5}^{\mathrm{2}} +\mathrm{4}\sqrt{\mathrm{3}}×\mathrm{6}}}=\frac{\mathrm{4}\sqrt{\mathrm{6}}}{\:\sqrt{\mathrm{50}+\mathrm{24}\sqrt{\mathrm{3}}}} \\ $$

Commented by Rasheed.Sindhi last updated on 15/Jan/22

$$ \\ $$

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