Question 221417


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,-0.2,2),
locate(1,2,8)
)}}}



To find the unknown length, we need to use the Pythagorean Theorem.



Remember, the Pythagorean Theorem is {{{a^2+b^2=c^2}}} where "a" and "b" are the legs of a triangle and "c" is the hypotenuse.



Since the legs are {{{x}}} and {{{2}}} this means that {{{a=x}}} and {{{b=2}}}


   

Also, since the hypotenuse is {{{8}}}, this means that {{{c=8}}}.



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



{{{x^2+2^2=8^2}}} Plug in {{{a=x}}}, {{{b=2}}}, {{{c=8}}} 



{{{x^2+4=8^2}}} Square {{{2}}} to get {{{4}}}.



{{{x^2+4=64}}} Square {{{8}}} to get {{{64}}}.



{{{x^2=64-4}}} Subtract {{{4}}} from both sides.



{{{x^2=60}}} Combine like terms.



{{{x=sqrt(60)}}} Take the square root of both sides. Note: only the positive square root is considered (since a negative length doesn't make sense).



{{{x=2*sqrt(15)}}} Simplify the square root.



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Answer:



So the solution is {{{x=2*sqrt(15)}}} which approximates to {{{x=7.746}}}.



So the exact length is {{{2*sqrt(15)}}} units while the approximate length is {{{7.746}}} units.