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 13434 by Nayon last updated on 20/May/17

ΔABC∼ΔDEF ΔABC=25 ΔDEF=35 ((ΣMadian^2 of ΔABC)/(ΣMadian^2 of ΔDEF))=?

$$\Delta{ABC}\sim\Delta{DEF}\: \\ $$$$\Delta{ABC}=\mathrm{25} \\ $$$$\Delta{DEF}=\mathrm{35} \\ $$$$\frac{\Sigma{Madian}^{\mathrm{2}} {of}\:\Delta{ABC}}{\Sigma{Madian}^{\mathrm{2}} \:{of}\:\Delta{DEF}}=? \\ $$

Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 25/May/17

Σm^2 =((2(b^2 +c^2 )−a^2 )/4)+((2(a^2 +c^2 )−b^2 )/4)+((2(a^2 +b^2 )−c^2 )/4)= =(3/4)(a^2 +b^2 +c^2 )=(3/4)Σa^2 ΔABC∽ΔDEF⇒(1/k^2 )=(S_(ABC) /S_(DEF) )=((25)/(35))=(5/7) (a/a^′ )=(b/b^′ )=(c/c^′ )=(1/k)=(√(5/7)) ⇒((Σm_(ABC) ^2 )/(Σm_(DEF) ^2 ))=((Σa^2 )/(Σa^′^2 ))=((Σa^2 )/(k^2 Σa^2 ))=(1/k^2 )=(5/7).■

$$\Sigma{m}^{\mathrm{2}} =\frac{\mathrm{2}\left({b}^{\mathrm{2}} +{c}^{\mathrm{2}} \right)−{a}^{\mathrm{2}} }{\mathrm{4}}+\frac{\mathrm{2}\left({a}^{\mathrm{2}} +{c}^{\mathrm{2}} \right)−{b}^{\mathrm{2}} }{\mathrm{4}}+\frac{\mathrm{2}\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)−{c}^{\mathrm{2}} }{\mathrm{4}}= \\ $$$$=\frac{\mathrm{3}}{\mathrm{4}}\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} \right)=\frac{\mathrm{3}}{\mathrm{4}}\Sigma{a}^{\mathrm{2}} \\ $$$$\Delta{ABC}\backsim\Delta{DEF}\Rightarrow\frac{\mathrm{1}}{{k}^{\mathrm{2}} }=\frac{{S}_{{ABC}} }{{S}_{{DEF}} }=\frac{\mathrm{25}}{\mathrm{35}}=\frac{\mathrm{5}}{\mathrm{7}} \\ $$$$\frac{{a}}{{a}^{'} }=\frac{{b}}{{b}^{'} }=\frac{{c}}{{c}^{'} }=\frac{\mathrm{1}}{{k}}=\sqrt{\frac{\mathrm{5}}{\mathrm{7}}} \\ $$$$\Rightarrow\frac{\Sigma{m}_{{ABC}} ^{\mathrm{2}} }{\Sigma{m}_{{DEF}} ^{\mathrm{2}} }=\frac{\Sigma{a}^{\mathrm{2}} }{\Sigma{a}^{'^{\mathrm{2}} } }=\frac{\Sigma{a}^{\mathrm{2}} }{{k}^{\mathrm{2}} \Sigma{a}^{\mathrm{2}} }=\frac{\mathrm{1}}{{k}^{\mathrm{2}} }=\frac{\mathrm{5}}{\mathrm{7}}.\blacksquare \\ $$

Answered by mrW1 last updated on 20/May/17

((ΣMadian^2 of ΔABC)/(ΣMadian^2 of ΔDEF))=(((side length of ΔABC)/(side length of ΔDEF)))^2 ((area of ΔABC)/(area of ΔDEF))=(((side length of ΔABC)/(side length of ΔDEF)))^2 ⇒((ΣMadian^2 of ΔABC)/(ΣMadian^2 of ΔDEF))=((area of ΔABC)/(area of ΔDEF))=((25)/(35))=(5/7)

$$\frac{\Sigma{Madian}^{\mathrm{2}} {of}\:\Delta{ABC}}{\Sigma{Madian}^{\mathrm{2}} \:{of}\:\Delta{DEF}}=\left(\frac{{side}\:{length}\:{of}\:\Delta{ABC}}{{side}\:{length}\:{of}\:\Delta{DEF}}\right)^{\mathrm{2}} \\ $$$$\frac{{area}\:{of}\:\Delta{ABC}}{{area}\:{of}\:\Delta{DEF}}=\left(\frac{{side}\:{length}\:{of}\:\Delta{ABC}}{{side}\:{length}\:{of}\:\Delta{DEF}}\right)^{\mathrm{2}} \\ $$$$\Rightarrow\frac{\Sigma{Madian}^{\mathrm{2}} {of}\:\Delta{ABC}}{\Sigma{Madian}^{\mathrm{2}} \:{of}\:\Delta{DEF}}=\frac{{area}\:{of}\:\Delta{ABC}}{{area}\:{of}\:\Delta{DEF}}=\frac{\mathrm{25}}{\mathrm{35}}=\frac{\mathrm{5}}{\mathrm{7}} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com