In the xy-plane, point R (2,3) and point S (5,6) are two vertices of triangle RST. If the sum of the slope of the sides of the triangle is 1, why can't angle T be a right angle?
Thanks in advance!
Edit: Hmm ... this seemed shorter in my head. Please bear the long-windedness of this post ...
Let the point be arbitrarily labeled .
Now if you form , it is essentially composed of 3 line segments:
Let be the respective slopes of .
We have: ..... .....
Now, we're told that the sums of the slopes equal 1, i.e. . Since , we have:
Also, note that is the same thing as saying that . If you remember, in order for two line segments to be perpendicular to one another, their slope one of line segment must be the negative reciprocal of the other.
So in order for , we must have:
Equate the red and blue to get:
Notice that they're reciprocals of each other. The only value that is equal to its own reciprocal is 1 and -1.
So either: .....or.....
But we're told that the sum of the slopes is equal 1 (from above in magenta):
Do you see how this can never happen?
In the xy-plane, point and point are two vertices of
If the sum of the slopes of the sides of the triangle is 1, why can't be a right angle?Code:| | S(5,6) | o | * * m2 | * * | * o T(x,y) | * * | * * m1 | o | R(2,3) | - + - - - - - - - - - - - - - |
Sum of the slopes is 1: . .
If , then: . .
Substitute  into : .
If is parallel to (coincides with)
. . and we don't have a triangle.
(In other words: . )
. . and a triangle cannot have two right angles.