Question 253715: Determine whether the given coordinates are the vertices of a triangle.
1. A(3,5), B(4,7), C(7,6)
2. S(6,5), T(8,3), U(12,-1)
3. H(-8,4), I(-4,2), J(4,-2)
4. D(1,-5), E(-3,0), F(-1,0)
5. R(1,3), S(4,0), T(10,-6)
6. W(2,6), X(1,6), Y(4,2)
7. P(-3,2), L(1,1), M(9,-1)
8. B(1,1), C(6,5), D(4,-1)
Found 2 solutions by richwmiller, Edwin McCravy:
Answer by richwmiller(17219)   (Show Source):
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Answer by Edwin McCravy(20056)  (Show Source):
You can put this solution on YOUR website!
I'll just do 1 of each type, number 1 and 3. The others
are done exactly like one of those. 1 forms a triangle and 3 doesn't.
1. A(3,5), B(4,7), C(7,6)
Yes they do. How to tell using algebra and without looking at a graph:
Three points either form a triangle or they form a line (said
to be "co-linear")
You find the slopes of the segments joining any two pairs of the
three points. If they are different, they don't form a straight
line, and thus they form a triangle. If they have the same slope
they form a straight line, and do not form a triangle.
So we find the slopes of AB and AC. (We could also have found the
slope of BC, but it is only necessary to find the slopes of two
of the segments:
 , 
They are not the same so the points do not lie in a straight line, and
therefore they form a triangle.
3. H(-8,4), I(-4,2), J(4,-2)
No they don't. They form a straight line. How to tell using algebra
and without looking at a graph:
Three points either form a triangle or they form a line (said
to be "co-linear"). These are co-linear.
You find the slopes of the segments joining any two pairs of the
three points. If they are different, they don't form a straight
line, and thus they form a triangle. If they have the same slope
they form a straight line, and do not form a triangle.
So we find the slopes of HI and IJ. (We could also have found the
slope of HJ, but it is only necessary to find the slopes of two
of the segments:
 , 
They are the same so the points lie in a straight line, and
therefore they do not form a triangle.
All the others are like one of these. You do the rest of them by
yourself.
Edwin
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