I dont get this SAT Math Question?

Question

I found this question in my SAT book and i got it wrong but i don't understand why. The explanation in the answer key doesn't make sense to me either.

Here's the question:

In the xy-plane, point R (2,3) and point S (5,6) are two vertices of triangle RST. If the sum of the slopes of the sides of the triangle is 1, which of the following angles could be a right angle?

I. angle R

II. angle S

III. angle T

(A) none

(B) I only

(C) III only

(D) I and II only

(E) I, II, and III

The answer key says it's (A) but i think it's (C)

If T is a right angle at (5,3), the sum of the slopes will still be 1, right?

I'm not sure and I'm eager to find the right answer.

Answer 1

The slope of segment RS is (6 - 3)/(5 - 2) = 3/3 = 1.  Given that, if R or S is the right angle, then the slope of the perpendicular side will have to be -1.  (Remember - the slopes of perpendicular lines are the negative reciprocals of each other.)  Since 1 + -1 = 0, in order for the slopes of all three segments to add to 1, the slope of the third side would have to be 1, making it parallel to RS and therefore incapable of forming a right triangle.

That leaves us with T.  If there is a right angle at T, then the slopes of RT and ST would have to be the negative reciprocals of each other.  Since the slope of RS is 1, it means the slopes of RT and ST would have to sum to 0.  So, let the slope of RT be m.  The slope of ST would have to be -1/m, and

m + (-1/m) = 0

(m^2 - 1)/m = 0

m^2 - 1 = 0

So m must be plus or minus 1.  However, if m is +1, then that segment is parallel to RS, and so we can't form a triangle with it.  If m is -1, then that segment is perpendicular to RS, and we already know that can't be the case.  It is, therefore, impossible for the right angle to be at T, and that was the last of our possibilities.  It must be A.

Edit:  If you see an answer with a link to answers-video or something along those lines, DO NOT click on it.  It's a malware distribution site, put there by a bot.  I've reported it, but it may take a bit for it to be deleted.

Answer 2

whoa, my brain hurts now.....