Question

which coordinate for points a and b would help prove that lines ab and ab are perpendicular? a(-m, p) a b b(w, z) \bigcirc a: (p, m) and b: (z, w) \bigcirc a: (p, m) and b: (z, -w) \bigcirc a: (p, -m) and b: (z, w) \bigcirc a: (p, -m) and b: (z, -w)

Explanation:

Step1: Find slope of line AB

Slope formula: $m_{AB} = \frac{z - p}{w - (-m)} = \frac{z - p}{w + m}$

Step2: Recall perpendicular slope rule

Perpendicular lines have slopes that are negative reciprocals: $m_{A'B'} = -\frac{w + m}{z - p}$

Step3: Test option D slopes

For A'$(p, -m)$ and B'$(z, -w)$:
$m_{A'B'} = \frac{-w - (-m)}{z - p} = \frac{m - w}{z - p} = -\frac{w - m}{z - p}$
Wait, correct reciprocal check: The slope of AB is $\frac{z-p}{w+m}$, so negative reciprocal is $-\frac{w+m}{z-p}$. Now, if we take A'$(p, -m)$ and B'$(z, -w)$:
$m_{A'B'} = \frac{-w - (-m)}{z - p} = \frac{m - w}{z - p}$. But another way: For perpendicular lines, if we have a point $(a,b)$, a perpendicular transformation swaps coordinates and negates one. For A$(-m,p)$, perpendicular point is $(p, m)$? No, wait, the product of slopes must be -1.
Let's compute product for option D:
$m_{AB} \times m_{A'B'} = \frac{z-p}{w+m} \times \frac{-w + m}{z-p} = \frac{m - w}{m + w}$? No, wait, correct option is B? No, wait let's do it properly.
Wait, let's take A$(-m,p)$, B$(w,z)$. Slope of AB is $\frac{z - p}{w + m}$.
For two lines to be perpendicular, $m_1 \times m_2 = -1$.
Let's test option B: A'$(p,m)$, B'$(z,-w)$
Slope of A'B' is $\frac{-w - m}{z - p} = -\frac{w + m}{z - p}$
Now multiply slopes: $\frac{z-p}{w+m} \times (-\frac{w+m}{z-p}) = -1$. This satisfies the perpendicular condition.
Wait, my earlier step was wrong. Let's correct:

Step1: Calculate slope of AB

$m_{AB} = \frac{z - p}{w - (-m)} = \frac{z - p}{w + m}$

Step2: Test option B slope

$m_{A'B'} = \frac{-w - m}{z - p} = -\frac{w + m}{z - p}$

Step3: Verify product is -1

$m_{AB} \times m_{A'B'} = \frac{z-p}{w+m} \times (-\frac{w+m}{z-p}) = -1$
This confirms perpendicularity.

Answer:

B. A': (p, m) and B': (z, -w)