Final answer:
Jessica must calculate the slope of the main hallway, the slope of a perpendicular hallway, the distance between the two hallways, and the midpoint of the main hallway. By using algebraic formulas, she will be able to determine these elements to create a precise map.
Step-by-step explanation:
Jessica is creating a map and working with the concepts of slope and distances in a coordinate plane, which is a fundamental topic in algebra.
Part A:
To find the slope of the main hallway, we use the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substituting the given points (−5, 9) and (4, 6) into the formula gives us \( m = \frac{6 - 9}{4 - (-5)} = \frac{-3}{9} = -\frac{1}{3} \). Thus, the slope of the main hallway is -1/3.
Part B:
The perpendicular slope to the main hallway can be found by taking the negative reciprocal of the main hallway's slope. Given that the slope of the main hallway is -1/3, the perpendicular slope would be 3.
Part C:
Since the two hallways are perpendicular, the distance between them at the point (−4, 3) can be considered as the vertical or horizontal distance from this point to the main hallway, depending on their relative positions on the map. This cannot be determined without additional information.
Part D:
To find the midpoint of the main hallway, we average the x-coordinates and y-coordinates of the given endpoints. The midpoint is \( (\frac{-5 + 4}{2}, \frac{9 + 6}{2}) \), which simplifies to (−0.5, 7.5), so the midpoint is (−0.5, 7.5).