Question and Answers Forum

All Questions      Topic List

Geometry Questions

Previous in All Question      Next in All Question      

Previous in Geometry      Next in Geometry      

Question Number 185828 by normans last updated on 28/Jan/23

Commented by mr W last updated on 28/Jan/23

(x+y+z)_(min) =(√((a^2 +b^2 +c^2 +4(√3)Δ)/2)) Δ=area of triangle ABC cos B=((2^2 +4^2 −(2(√3))^2 )/(2×2×4))=(1/2) ⇒sin B=((√3)/2) Δ=((2×4×(√3))/(2×2))=2(√3) (x+y+z)_(min) =(√((2^2 +4^2 +(2(√3))^2 +4(√3)×2(√3))/2)) =2(√7)

$$\left({x}+{y}+{z}\right)_{{min}} =\sqrt{\frac{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +\mathrm{4}\sqrt{\mathrm{3}}\Delta}{\mathrm{2}}} \\ $$$$\Delta={area}\:{of}\:{triangle}\:{ABC} \\ $$$$\mathrm{cos}\:{B}=\frac{\mathrm{2}^{\mathrm{2}} +\mathrm{4}^{\mathrm{2}} −\left(\mathrm{2}\sqrt{\mathrm{3}}\right)^{\mathrm{2}} }{\mathrm{2}×\mathrm{2}×\mathrm{4}}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\Rightarrow\mathrm{sin}\:{B}=\frac{\sqrt{\mathrm{3}}}{\mathrm{2}} \\ $$$$\Delta=\frac{\mathrm{2}×\mathrm{4}×\sqrt{\mathrm{3}}}{\mathrm{2}×\mathrm{2}}=\mathrm{2}\sqrt{\mathrm{3}} \\ $$$$\left({x}+{y}+{z}\right)_{{min}} =\sqrt{\frac{\mathrm{2}^{\mathrm{2}} +\mathrm{4}^{\mathrm{2}} +\left(\mathrm{2}\sqrt{\mathrm{3}}\right)^{\mathrm{2}} +\mathrm{4}\sqrt{\mathrm{3}}×\mathrm{2}\sqrt{\mathrm{3}}}{\mathrm{2}}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{2}\sqrt{\mathrm{7}} \\ $$

Commented by mr W last updated on 28/Jan/23

see Q164174

$${see}\:{Q}\mathrm{164174} \\ $$

Commented by normans last updated on 28/Jan/23

nice sir

$${nice}\:{sir} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com