Question 253715
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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.
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1. A(3,5), B(4,7), C(7,6)
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{{{drawing(400,272.73,-9,13,-7,8, graph(400,272.73,-9,13,-7,8),
triangle(3,5,4,7,7,6),locate(3,5,A), locate(4,7,B),locate(7,6,C) )}}}

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:

{{{m[AB]=(7-5)/(4-3)=2/1=2}}}, {{{m[AC]=(6-5)/(7-3)=1/4}}}

They are not the same so the points do not lie in a straight line, and
therefore they form a triangle.
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3. H(-8,4), I(-4,2), J(4,-2)
<pre><font size = 4 color = "indigo"><b>
{{{drawing(400,272.73,-9,13,-7,8, graph(400,272.73,-9,13,-7,8),
triangle(-8,4,-4,2,4,-2),locate(-8,4,H), locate(-4,2,I),locate(4,-2,J) )}}}

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:

{{{m[HI]=(2-4)/(-4-(-8))=(-2)/(-4+8)=(-2)/4=-1/2}}}, {{{m[IJ]=(-2-2)/(4-(-4))=(-4)/(4+4)=(-4)/8=-1/2}}}

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</pre>