BB Math Pg. 311 #15

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1

<p>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. R
II. S
III. T
A. None
B. I only
C. III only
D. I and II only
E. I, II, and III</p>

<p>I think it's C, because the slope of RS is already 1, but if point T was at (5,3), this would form a right triangle with the hypotenuse having the slope of one and the other two legs would have a slope of undefined and 0. This would add up to 1. However, the answer is A. The only thing I can see is that they have a problem with adding an undefined slope.</p>

2

<p>If a right angle is drawn from either (2,3) or (5,6) it will create a side with slope of -1. However, in order for it to add up to 1, the last side has to have a slope of 1. If this happens, then two of the sides of the triangle have to be parallel, something that cannot happen.</p>

3

<p>True, rcchen, but that only applies to drawing a right angle from R and S; T wouldn't have that problem.</p>

<p>I find the question puzzling. The undefined slope is the only potential problem I can think of.</p>

4

<p>Does anyone have insight on this?</p>

5

<p>rcchen correctly ruled out angles R and S, and the problem for angle T is undefined sloped as 112358 guessed. The length between R and S is sqrt(3^2+3^2) = sqrt(18). To make T a right angle, you need a point somewhere on the line with slope -1 that goes through the midpoint of R and S. A point on this line, call it line l, just off of line RS will have an angle of close to 180 when the triangle is made, while a point "far away" will have an angle close to zero, so there is a point T on l such that angle T is 90 in RST. </p>

<p>When you make this, though, sides RT and ST have the same length, and they are the two sides of the right triangle whose hypotenuse is RS which has length sqrt(18), so each side has length 3, as before when the slope was taken since that was (6-3)/(5-2) = 3/3. If you find the rise/run using the bottom side of the line RS, then you will see that RT, ST, and the two sides for the rise over run on the bottom form a square. Two of the sides of the square have undefined slope and the other two have slope zero. RT has undefined slope, ST has slope zero. 1+0+undefined = undefined, so the sum of the slopes cannot be 1 in this right triangle.</p>