Question 181935


We basically have this triangle set up:



{{{drawing(500,500,-0.5,2,-0.5,3.2,
line(0,0,0,3),
line(0,3,2,0),
line(2,0,0,0),
locate(-0.2,1.5,x),
locate(1,-0.2,x-7),
locate(1,2,13)
)}}}



Since the legs are {{{x}}} and {{{x-7}}} this means that {{{a=x}}} and {{{b=x-7}}}


   

Also, since the hypotenuse is {{{13}}}, this means that {{{c=13}}}.



{{{a^2+b^2=c^2}}} Start with the Pythagorean theorem.



{{{x^2+(x-7)^2=13^2}}} Plug in {{{a=x}}}, {{{b=x-7}}}, {{{c=13}}} 



{{{x^2+(x-7)^2=169}}} Square {{{13}}} to get {{{169}}}.



{{{x^2+x^2-14x+49=169}}} FOIL



{{{x^2+x^2-14x+49-169=0}}} Subtract 169 from both sides.



{{{2x^2-14x-120=0}}} Combine like terms.





Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=2}}}, {{{b=-14}}}, and {{{c=-120}}}



Let's use the quadratic formula to solve for x



{{{x = (-b +- sqrt( b^2-4ac ))/(2a)}}} Start with the quadratic formula



{{{x = (-(-14) +- sqrt( (-14)^2-4(2)(-120) ))/(2(2))}}} Plug in  {{{a=2}}}, {{{b=-14}}}, and {{{c=-120}}}



{{{x = (14 +- sqrt( (-14)^2-4(2)(-120) ))/(2(2))}}} Negate {{{-14}}} to get {{{14}}}. 



{{{x = (14 +- sqrt( 196-4(2)(-120) ))/(2(2))}}} Square {{{-14}}} to get {{{196}}}. 



{{{x = (14 +- sqrt( 196--960 ))/(2(2))}}} Multiply {{{4(2)(-120)}}} to get {{{-960}}}



{{{x = (14 +- sqrt( 196+960 ))/(2(2))}}} Rewrite {{{sqrt(196--960)}}} as {{{sqrt(196+960)}}}



{{{x = (14 +- sqrt( 1156 ))/(2(2))}}} Add {{{196}}} to {{{960}}} to get {{{1156}}}



{{{x = (14 +- sqrt( 1156 ))/(4)}}} Multiply {{{2}}} and {{{2}}} to get {{{4}}}. 



{{{x = (14 +- 34)/(4)}}} Take the square root of {{{1156}}} to get {{{34}}}. 



{{{x = (14 + 34)/(4)}}} or {{{x = (14 - 34)/(4)}}} Break up the expression. 



{{{x = (48)/(4)}}} or {{{x =  (-20)/(4)}}} Combine like terms. 



{{{x = 12}}} or {{{x = -5}}} Simplify. 



So the possible answers are {{{x = 12}}} or {{{x = -5}}} 

  


However, a negative length is NOT possible. So {{{x = -5}}} is NOT a solution



So the only answer is {{{x = 12}}} 






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Answer:



So the solution is {{{x = 12}}}



This means that the other leg is {{{12-7=5}}}



So the two legs are: 12 and 5 units