Question 94571
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.
<pre><font size = 5 color = "indigo"><b>
First plot the points and draw the triangle so
you can tell which one looks most like the 
vertex of a right angle.

{{{drawing(400,375,-4,6,-1,9,

triangle(-3,1,2,-1,6,9), graph(400,375,-4,6,-1,9),
locate(-4,1,"(-3,1)"), locate(2.1,-.7,"(2,-1)"), locate(4.7,9,"(6,9)")
 )}}}

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 = <font face = "symbol">Ö</font>(x<sub>2</sub>-x<sub>1</sub>)² + (y<sub>2</sub>-y<sub>1</sub>)²

with (x<sub>1</sub>,y<sub>1</sub>) = (-3,1) and (x<sub>2</sub>,y<sub>2</sub>) = (6,9)  

     __________________ 
D = <font face = "symbol">Ö</font>(6-(-3))² + (9-1)²
     ___________
D = <font face = "symbol">Ö</font>(6+3)² + 8²
     _______
D = <font face = "symbol">Ö</font>9² + 8²
     _______ 
D = <font face = "symbol">Ö</font>81 + 64
     ___
D = <font face = "symbol">Ö</font>145 = hypotenuse

Now we find the shorter leg, which connects 
the point (-3,1) to the point (2,-1)
     __________________ 
D = <font face = "symbol">Ö</font>(x<sub>2</sub>-x<sub>1</sub>)² + (y<sub>2</sub>-y<sub>1</sub>)²

with (x<sub>1</sub>,y<sub>1</sub>) = (-3,1) and (x<sub>2</sub>,y<sub>2</sub>) = (2,-1)  

     __________________ 
D = <font face = "symbol">Ö</font>(2-(-3))² + (-1-1)²
     ______________
D = <font face = "symbol">Ö</font>(2+3)² + (-2)²
     ______
D = <font face = "symbol">Ö</font>5² + 4
     ______ 
D = <font face = "symbol">Ö</font>25 + 4
     __
D = <font face = "symbol">Ö</font>29 = shorter leg

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

     __________________ 
D = <font face = "symbol">Ö</font>(x<sub>2</sub>-x<sub>1</sub>)² + (y<sub>2</sub>-y<sub>1</sub>)²

with (x<sub>1</sub>,y<sub>1</sub>) = (-2,1) and (x<sub>2</sub>,y<sub>2</sub>) = (6,9)  
<font face = "symbol">Ö</font>
     __________________ 
D = <font face = "symbol">Ö</font>(6-2)² + (9-(-1))²
     ___________
D = <font face = "symbol">Ö</font>4² + (9+1)²
     ________
D = <font face = "symbol">Ö</font>16 + 10²
     ________ 
D = <font face = "symbol">Ö</font>16 + 100
     ___
D = <font face = "symbol">Ö</font>116 = longer leg

So we see if this Pythagorean equation holds:
              ?
(hypotenuse)² = (shorter leg)² + (longer leg)²
        ___   ?   __      ___  
      (<font face = "symbol">Ö</font>145)² = (<font face = "symbol">Ö</font>29)² + <font face = "symbol">Ö</font>116)²
              ?  
          145 = 29 + 116
              <font face = "symbol">Ö</font>                
          145 = 145

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

Edwin</pre>