Question 146784
{{{P=2(L+W)}}} Start with the perimeter formula



{{{38=2(L+W)}}} Plug in {{{P=38}}}



{{{19=L+W}}} Divide both sides by 2. So {{{L+W=19}}}



{{{L^2+W^2=D^2}}} Now move onto the Pythagorean equation dealing with the diagonal and the side lengths



{{{L^2+W^2=15^2}}} Plug in {{{D=15}}}



{{{L^2+W^2=225}}} Square 15 to get 225



{{{L^2+W^2+2LW-2LW=225}}} Now add <b>and</b> subtract the quantity {{{2LW}}} on the left side. Adding <b>and</b> subtracting the same quantity does not change the equation.



{{{(L^2+2LW+W^2)-2LW=225}}} Group the terms



{{{(L+W)^2-2LW=225}}} Factor {{{L^2+2LW+W^2}}} to get {{{(L+W)^2}}}



{{{(19)^2-2LW=225}}} Plug in {{{L+W=19}}}



{{{361-2LW=225}}} Square 19 to get 361



{{{361-2A=225}}} Now replace {{{LW}}} with {{{A}}}. Remember, {{{A=LW}}}



{{{-2A=225-361}}} Subtract {{{361}}} from both sides.



{{{-2A=-136}}} Combine like terms on the right side.



{{{A=(-136)/(-2)}}} Divide both sides by {{{-2}}} to isolate {{{A}}}.



{{{A=68}}} Reduce.



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Answer:


So the answer is {{{A=68}}} 



So given a diagonal of 15 cm and a perimeter of 38 cm, the area is 68 {{{cm^2}}}




Note: You can go another way and solve for L and W (which turn out to be {{{(19-sqrt(89))/2}}} and {{{(19+sqrt(89))/2}}}) and find the area by multiplying L and W