SOLUTION: Length of a Line Segment 20. The point A(x, 1) is 5 units from the point (2, 6). a) Find a possible value for x. b) Is this value the only solution? Explain. Thank you very muc

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: Length of a Line Segment 20. The point A(x, 1) is 5 units from the point (2, 6). a) Find a possible value for x. b) Is this value the only solution? Explain. Thank you very muc      Log On


   

Question 188716: Length of a Line Segment
20. The point A(x, 1) is 5 units from the point (2, 6).
a) Find a possible value for x.
b) Is this value the only solution? Explain.
Thank you very much!!! pleaseeeeee
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Remember, the distance between any two points and is

d=sqrt%28%28x%5B1%5D-x%5B2%5D%29%5E2%2B%28y%5B1%5D-y%5B2%5D%29%5E2%29


In this case, is the point (x,1) and is the point (2,6)



Now matching corresponding coordinates, we get x%5B1%5D=x, y%5B1%5D=1, x%5B2%5D=2, and y%5B2%5D=6



d=sqrt%28%28x%5B1%5D-x%5B2%5D%29%5E2%2B%28y%5B1%5D-y%5B2%5D%29%5E2%29 Start with the distance formula


5=sqrt%28%28x-2%29%5E2%2B%281-6%29%5E2%29 Plug in d=5 (the given distance) x%5B1%5D=x, y%5B1%5D=1, x%5B2%5D=2, and y%5B2%5D=6 (the given coordinates)


The goal now is to solve for "x":


5=sqrt%28%28x-2%29%5E2%2B%28-5%29%5E2%29 Combine like terms.


5=sqrt%28%28x-2%29%5E2%2B25%29 Square -5 to get 25


5%5E2=%28x-2%29%5E2%2B25 Square both sides (to eliminate the square root)


25=%28x-2%29%5E2%2B25 Square 5 to get 25


25-25=%28x-2%29%5E2 Subtract 25 from both sides.


0=%28x-2%29%5E2 Combine like terms.


%28x-2%29%5E2=0 Rearrange the equation


x-2=0%2B-sqrt%280%29 Take the square root of both sides.


x-2=sqrt%280%29 or x-2=-sqrt%280%29 Break up the "plus/minus" to form two equations.


x-2=0 or x-2=0 Take the square root of 0 to get 0.


x=2%2B0 or x=2-0 Add 2 to both sides.


x=2 or x=2 Combine like terms.


x=2 Remove the duplicate solution


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Answer:


So the only solution is x=2.



So the point (2,1) is 5 units from (2,6)


Note: if you plot the points on a coordinate plane and count the number of units one point is from another, you'll find that the two points are 5 units apart.