Question

instructions
choose the best answer. if necessary, use the paper you were given.
question
the coordinates of point a in the figure above are (p, r), where |p| > |r|. which of the following could be
the slope of ab?
-2
-1/2
2/3
5/2

Explanation:

Step1: Determine the signs of coordinates

Point \(A(p,r)\) is in the second - quadrant, so \(p\lt0\) and \(r\gt0\). Point \(B\) is in the fourth - quadrant. Let the coordinates of \(B\) be \((q,s)\), then \(q\gt0\) and \(s\lt0\).
The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For points \(A(p,r)\) and \(B(q,s)\), the slope \(m=\frac{s - r}{q - p}\).
Since \(s - r\lt0\) (because \(s\lt0\) and \(r\gt0\)) and \(q - p\gt0\) (because \(q\gt0\) and \(p\lt0\)), the slope \(m\lt0\). So we can eliminate options \(C\) (\(\frac{2}{3}\)) and \(D\) (\(\frac{5}{2}\)).

Step2: Analyze the magnitude of the slope

We know that \(|p|\gt|r|\). Let's assume \(A(-a,b)\) (\(a\gt0,b\gt0,a > b\)) and \(B(c,-d)\) (\(c\gt0,d\gt0\)). The slope \(m=\frac{-d - b}{c+ a}\).
If we assume \(A(-2,1)\) (since \(|p|\gt|r|\)), and assume \(B(2,-1)\) (for simplicity of calculation, just to check the magnitude). The slope \(m=\frac{-1 - 1}{2+2}=-\frac{1}{2}\).
If we assume \(A(-1,1)\) and \(B(1,-1)\), the slope \(m=-1\). If \(A(-3,1)\) and \(B(3,-1)\), the slope \(m =-\frac{1}{3}\).
We know that \(|p|\gt|r|\). Let \(A(p,r)\) and \(B\) be symmetric about the origin (a special case for slope calculation). The slope \(m=\frac{r - (-r)}{p-(-p)}=\frac{2r}{2p}=\frac{r}{p}\). Since \(|p|\gt|r|\), \(|\frac{r}{p}|\lt1\).

Answer:

\(-\frac{1}{2}\)