SOLUTION: The converse of the Pythagorean theorem is also a true statement: If the sum of the squares of the lengths of two sides of a triangle is is equal to the square of the length of the

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Question 94571This question is from textbook college algebra
: The converse of the Pythagorean theorem is also a true statement: If the sum of the squares of the lengths of two sides of a triangle is is equal to the square of the length of the third side, then the triangle is a right triangle. Use the distance formula and the Pythagorean theorem to determine whether the set of points could be vertices of a right triangle.
(-3,1),(2,-1) and (6,9)
How is this problem solved? Please show all the steps clearly. Thank you. This question is from textbook college algebra

Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
The converse of the Pythagorean theorem is also a true statement: If the sum of the squares of the lengths of two sides of a triangle is is equal to the square of the length of the third side, then the triangle is a right triangle. Use the distance formula and the Pythagorean theorem to determine whether the set of points could be vertices of a right triangle.
(-3,1),(2,-1) and (6,9)
How is this problem solved? Please show all the steps clearly. Thank you.

First plot the points and draw the triangle so
you can tell which one looks most like the 
vertex of a right angle.



Well, from looking at that picture the angle 
at (2,-1) looks the most like a right angle.  
So if this is really a right triangle, that 
would make the hypotenuse be the side connecting
the points (-3,1) and (6,9), so we use the 
formula for the distance between two points to 
calculate the hypotenuse:
     __________________ 
D = Ö(x2-x1)² + (y2-y1)²

with (x1,y1) = (-3,1) and (x2,y2) = (6,9)  

     __________________ 
D = Ö(6-(-3))² + (9-1)²
     ___________
D = Ö(6+3)² + 8²
     _______
D = Ö9² + 8²
     _______ 
D = Ö81 + 64
     ___
D = Ö145 = hypotenuse

Now we find the shorter leg, which connects 
the point (-3,1) to the point (2,-1)
     __________________ 
D = Ö(x2-x1)² + (y2-y1)²

with (x1,y1) = (-3,1) and (x2,y2) = (2,-1)  

     __________________ 
D = Ö(2-(-3))² + (-1-1)²
     ______________
D = Ö(2+3)² + (-2)²
     ______
D = Ö5² + 4
     ______ 
D = Ö25 + 4
     __
D = Ö29 = shorter leg

Now we find the longer leg, from 
(-2,1) to (6,9)

     __________________ 
D = Ö(x2-x1)² + (y2-y1)²

with (x1,y1) = (-2,1) and (x2,y2) = (6,9)  
Ö
     __________________ 
D = Ö(6-2)² + (9-(-1))²
     ___________
D = Ö4² + (9+1)²
     ________
D = Ö16 + 10²
     ________ 
D = Ö16 + 100
     ___
D = Ö116 = longer leg

So we see if this Pythagorean equation holds:
              ?
(hypotenuse)² = (shorter leg)² + (longer leg)²
        ___   ?   __      ___  
      (Ö145)² = (Ö29)² + Ö116)²
              ?  
          145 = 29 + 116
              Ö                
          145 = 145

So, yes, the Pythagorean equation holds so, 
it is a right triangle

Edwin