We are given a line $g$ in the Cartesian coordinate plane, where it intersects the x-axis at $(16, 0)$ and the y-axis at $(0, -12)$. Another line, $k$, is perpendicular to line $g$ and intersects line $g$ at the point $(0, -12)$. We are tasked with finding the x-intercept of line $k$, which means we need to find the point where line $k$ crosses the x-axis.

Step 1: Find the slope of line $g$

The slope $m$ of a line is given by: $m = \frac{y_2 - y_1}{x_2 - x_1}$ For line $g$, we know the coordinates of two points: $(0, -12)$ and $(16, 0)$. So, the slope of $g$ is: $m_g = \frac{0 - (-12)}{16 - 0} = \frac{12}{16} = \frac{3}{4}$

Step 2: Determine the slope of line $k$

Since line $k$ is perpendicular to line $g$, its slope will be the negative reciprocal of the slope of line $g$. The slope of $g$ is $\frac{3}{4}$, so the slope of $k$ is: $m_k = -\frac{4}{3}$

Step 3: Find the equation of line $k$

The general form of a line is: $y = mx + c$ We know the slope of line $k$ is $-\frac{4}{3}$, and it passes through the point $(0, -12)$. Substituting into the equation: $-12 = -\frac{4}{3}(0) + c$ This gives $c = -12$. Therefore, the equation of line $k$ is: $y = -\frac{4}{3}x - 12$

Step 4: Find the x-intercept of line $k$

The x-intercept occurs when $y = 0$. Substituting $y = 0$ into the equation of line $k$: $0 = -\frac{4}{3}x - 12$ Solve for $x$: $\frac{4}{3}x = -12$ Multiply both sides by $\frac{3}{4}$: $x = -12 \times \frac{3}{4} = -9$

Thus, the x-intercept of line $k$ is $(-9, 0)$.

Answer:

The correct answer is B. (-9, 0).

Would you like more details or have any further questions?

Here are 5 related questions for further practice:

1. How would the equation of line $g$ change if its slope were altered?
2. What would happen if line $k$ intersected line $g$ at a different point?
3. Can you determine the y-intercept of line $g$ directly from the diagram?
4. How do perpendicular lines behave geometrically in other types of coordinate systems?
5. What method would you use to find the point of intersection of two lines algebraically?

Tip: When working with perpendicular lines, always remember that their slopes are negative reciprocals of each other.