Question 1203533
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<font color=red size=4>Answer:</font> No, this is not a right triangle.


Explanation


Label the points A,B,C. The order of which doesn't matter.
A = (2,7)
B = (3,5)
C = (1,0)


I'll use the distance formula to compute how far it is from A to B. 
This will give us the length of segment AB.


A = (x1,y1) = (2,7) and B = (x2,y2) = (3,5)
d = distance from A to B = length of segment AB
{{{d = sqrt( (x1-x2)^2 + (y1-y2)^2 )}}}


{{{d = sqrt( (2-3)^2 + (7-5)^2 )}}}


{{{d = sqrt( (-1)^2 + (2)^2 )}}}


{{{d = sqrt( 1 + 4 )}}}


{{{d = sqrt( 5 )}}}


{{{d = 2.236068}}}


Segment AB is exactly {{{sqrt(5)}}} units long, which approximates to about 2.236068 units.


Use that same formula to find these other segment lengths:
{{{BC = sqrt(29) = 5.385165}}}
{{{AC = sqrt(50) = 5*sqrt(2) = 7.071068}}}
The decimal values are approximate.


Once we know the side lengths, we can plug them into the pythagorean theorem equation. 
If we get the same thing on both sides, then we conclude the triangle is a right triangle.


{{{(matrix(1,2,"first","leg"))^2+(matrix(1,2,"second","leg"))^2=(hypotenuse)^2}}}


{{{(AB)^2+(BC)^2=(AC)^2}}} The hypotenuse is <u><font color=red>always</font></u> the longest side. In this case, the longest side is AC = sqrt(50) = 7.071068 roughly.


{{{(sqrt(5))^2+(sqrt(29))^2=(sqrt(50))^2}}}


{{{5+29=50}}}


{{{34=50}}}


Because we ended up with a false equation, it means that {{{(AB)^2+(BC)^2=(AC)^2}}} is also false. 
Furthermore, we do NOT have a right triangle.


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Here's another approach that might be more simple depending on your viewpoint.


Once again we have these three given points
A = (2,7)
B = (3,5)
C = (1,0)


Let's compute the slope of line AB.


A = (x1,y1) = (2,7) and B = (x2,y2) = (3,5)
{{{m = slope = rise/run = matrix(1,3,"change","in","y")/matrix(1,3,"change","in","x")}}}


{{{m = (y[2] - y[1])/(x[2] - x[1])}}}


{{{m = (5 - 7)/(3 - 2)}}}


{{{m = (-2)/(1)}}}


{{{m = -2}}}
The slope of line AB is -2.


If you use the slope formula for the other sides, then:
slope of BC = 5/2 = 2.5
slope of AC = 7


Then multiply each possible slope pairing.
(slope AB)*(slope BC) = (-2)*(2.5) = -5
(slope AB)*(slope AC) = (-2)*(7) = -14
(slope BC)*(slope AC) = (2.5)*(7) = 17.5


None of the pairs multiply to -1. 
Therefore, none of the pairs of lines are perpendicular and we do NOT have a right triangle.


There might be other approaches. Feel free to explore other alternatives.
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