Question 785425: Show that the triangle whose vertices are A(4,3) B(6,-2) C(-11,3) is a right angled triangle.
Please help!!
Thanks
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website!  It is not a right triangle. There is a typo somewhere.
Or else the problem jus asked to FIND IF it was a right triangle.
To see if you have a right angle you look at the slopes of the lines involved.
If two lines form a right angle, they are perpendicular.
If two lines are perpendicular,
eihter one is horizontal and the other vertical (easy to see without calculations),
or the product of their slopes is -1.
If the triangle had a right angle, two of the 3 sides (AB, BC, and AC) would be perpendicular to each other.
AC is clearly horizontal, along the line y=3 (slope=0).
A line perpendicular to AC whould be vertical, with a fixed value for x.
That vewrtical line would have no slope (slope would be undefined.
I see that here is no vertical side in that triangle, because the x-coordinates of the 3 points are all different.
If B had the same x=4 as A (as in B(4,-2) for example), AB would be vertical, and you would have a right angle at A (AB and AC would be perpendicular).
If B had the same x=-11 as C (as in B(-11,-2) for example), BC would be vertical, and you would have a right angle at C (BC and AC would be perpendicular).
That means the right angle does not involve side AC. It is not at A or at C, so it must be at vertex B.
We must prove that lines AB and BC are perpendicular.
Neither of those lines is vertical, so they both have a slope.
AB and BC would be perpendicular if and only if the product of their slopes were -1.
slope of AB=
slope of BC=
 so AB and BC are not perpendicular. There is no right angle at B.
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