Question

question 10 (multiple - choice, 1 point) (04.02 mc) segment ab is on the line y - 9=-4(x + 1), and segment cd is on the line y - 6 = 1/4(x - 3). which statement proves the relationship of segments ab and cd? o they are parallel because they have the same slope of - 4. o they are perpendicular because they have slopes that are opposite reciprocals of - 4 and 1/4. o they are parallel because they have the same slope of 4. o they are perpendicular because they have slopes that are opposite reciprocals of 4 and - 1/4.

Explanation:

Step1: Rewrite equations in slope - intercept form

The point - slope form of a line is $y - y_1=m(x - x_1)$. For line segment $AB$ with equation $y - 9=-4(x + 1)$, we can rewrite it as $y=-4x+5$ (using distributive property: $y-9=-4x - 4$, then $y=-4x+5$), so the slope $m_{AB}=-4$. For line segment $CD$ with equation $y - 6=\frac{1}{4}(x - 3)$, we rewrite it as $y=\frac{1}{4}x+\frac{21}{4}$ (using distributive property: $y - 6=\frac{1}{4}x-\frac{3}{4}$, then $y=\frac{1}{4}x-\frac{3}{4}+6=\frac{1}{4}x+\frac{- 3 + 24}{4}=\frac{1}{4}x+\frac{21}{4}$), so the slope $m_{CD}=\frac{1}{4}$.

Step2: Recall perpendicular - slope relationship

Two lines are perpendicular if the product of their slopes is $- 1$. The product of $-4$ and $\frac{1}{4}$ is $-4\times\frac{1}{4}=-1$. Also, perpendicular lines have slopes that are opposite reciprocals of each other. The opposite reciprocal of $-4$ is $\frac{1}{4}$.

Answer:

They are perpendicular because they have slopes that are opposite reciprocals of $-4$ and $\frac{1}{4}$