Question 922349

The diagonal {{{d}}} of a rectangle has a length {{{50}}}: then

{{{d=50}}}


If its legs {{{a}}} and {{{b}}} are in the ratio {{{3:4}}} , then we have:

{{{a:b=3:4}}} ...solve for {{{a}}}

{{{4a=3b}}}

{{{a=3b/4}}}

{{{a=(3/4)b}}}



what is its perimeter? 

the diagonal of a rectangle along with the legs {{{a}}} and {{{b}}} forms a right angle triangle; so, usu Pythagorean theorem to find the length of the legs

{{{d^2=a^2+b^2}}} ....substitute {{{a=(3/4)b}}} and {{{d=50}}}

{{{50^2=((3/4)b)^2+b^2}}}


{{{2500=9b^2/16+b^2}}} ...solve for {{{b}}}


{{{2500=9b^2/16+16b^2/16}}}


{{{2500=25b^2/16}}}


{{{2500*16=25b^2}}}


{{{(2500*16)/25=b^2}}}


{{{(cross(2500)100*16)/cross(25)=b^2}}}


{{{100*16=b^2}}}


{{{1600=b^2}}}


{{{b=sqrt(1600)}}}


{{{highlight(b=40)}}} or {{{highlight(b=-40)}}}

since we have a length, we don't need negative solution;

so, {{{highlight(b=40)}}}

now go back to {{{d^2=a^2+b^2}}}, plug in {{{d=50}}} and {{{b=40}}} and calculate {{{a}}}

{{{50^2=a^2+40^2}}}

{{{2500=a^2+1600}}}

{{{2500-1600=a^2}}}

{{{900=a^2}}}

{{{sqrt(900)=a}}}

{{{highlight(a=30)}}}

now we know the length of the sides and we can calculate perimeter {{{P}}}:

{{{P=2a+2b}}}

{{{P=2*30+2*40}}}

{{{P=60+80}}}

{{{highlight(P=140)}}}