Question

ntifying the criteria for perpendicular lines with the same y - intercept
what are the necessary criteria for a line to be perpendicular to the given line and have the same y - intercept?
the slope is - 3/2 and contains the point (0, 2).
the slope is - 2/3 and contains the point (0, - 2).
the slope is 2/3 and contains the point(0, 2).
the slope is - 3/2 and contains the point (0, - 2).

Explanation:

Step1: Recall slope - relationship of perpendicular lines

If two lines with slopes $m_1$ and $m_2$ are perpendicular, then $m_1\times m_2=-1$. Let the slope of the given line be $\frac{3}{2}$. Then the slope of the perpendicular line $m$ satisfies $\frac{3}{2}\times m=-1$, so $m =-\frac{2}{3}$.

Step2: Identify y - intercept

The y - intercept is the value of $y$ when $x = 0$. We want the perpendicular line to have the same y - intercept as the given line. From the options, we need a point of the form $(0,b)$. The y - intercept of the given line is not clearly shown in the graph, but among the options, the point $(0, - 2)$ is part of the correct option for the perpendicular line with slope $-\frac{2}{3}$.

Answer:

The slope is $-\frac{2}{3}$ and contains the point $(0, - 2)$