Menu Close

Question-82042




Question Number 82042 by ajfour last updated on 17/Feb/20
Commented by ajfour last updated on 17/Feb/20
Find ratio of side lengths of  outer equilateral △KLM to  that of inner equilateral △PQR.
$${Find}\:{ratio}\:{of}\:{side}\:{lengths}\:{of} \\ $$$${outer}\:{equilateral}\:\bigtriangleup{KLM}\:{to} \\ $$$${that}\:{of}\:{inner}\:{equilateral}\:\bigtriangleup{PQR}. \\ $$
Commented by mr W last updated on 19/Feb/20
the max. outer equilateral is  (see Q60313)  l_(max) =(√(((2(a^2 +b^2 +c^2 ))/3)+((8Δ)/( (√3)))))    the min. inner equilateral is  (see Q82131)  l_(min) =((2(√2)Δ)/( (√(a^2 +b^2 +c^2 +4(√3)Δ))))
$${the}\:{max}.\:{outer}\:{equilateral}\:{is} \\ $$$$\left({see}\:{Q}\mathrm{60313}\right) \\ $$$${l}_{{max}} =\sqrt{\frac{\mathrm{2}\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} \right)}{\mathrm{3}}+\frac{\mathrm{8}\Delta}{\:\sqrt{\mathrm{3}}}} \\ $$$$ \\ $$$${the}\:{min}.\:{inner}\:{equilateral}\:{is} \\ $$$$\left({see}\:{Q}\mathrm{82131}\right) \\ $$$${l}_{{min}} =\frac{\mathrm{2}\sqrt{\mathrm{2}}\Delta}{\:\sqrt{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +\mathrm{4}\sqrt{\mathrm{3}}\Delta}} \\ $$
Commented by ajfour last updated on 17/Feb/20
In terms of a,b,c  Sir, lets   determine the two possible  ratios..
$${In}\:{terms}\:{of}\:{a},{b},{c}\:\:{Sir},\:{lets}\: \\ $$$${determine}\:{the}\:{two}\:{possible} \\ $$$${ratios}.. \\ $$
Commented by ajfour last updated on 25/Feb/20
Thanks sir, its a long story cut  short, up here.
$$\mathrm{Thanks}\:\mathrm{sir},\:\mathrm{its}\:\mathrm{a}\:\mathrm{long}\:\mathrm{story}\:\mathrm{cut} \\ $$$$\mathrm{short},\:\mathrm{up}\:\mathrm{here}. \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *