Final answer:
The equation of the line perpendicular to the one through (9, 10) and (3, -2) that passes through the line's x-intercept is not present in the given options, as the correct equation should be y = -1/2x + 7, based on the provided steps of finding the negative reciprocal of the original line's slope and determining the y-intercept of the perpendicular line.
The correct option is not given.
Step-by-step explanation:
The goal is to find the equation of a line that is perpendicular to the line passing through the points (9, 10) and (3, -2) and that goes through the x-intercept of this line. To start, we calculate the slope of the given line, which is ∆y / ∆x = (10 - (-2)) / (9 - 3) = 12 / 6 = 2.
The slope of the perpendicular line will be the negative reciprocal, which is -1 / 2. We know the original line crosses the x-axis when y equals zero. So, for the line y = 2x + b, finding the x-intercept gives us 0 = 2x + b. Solving for x when y = 0 gives us x = -b/2.
As the x-intercept is a point on both lines, we can use it to find the equation of the perpendicular line. If x = -b/2, then using the slope -1/2, the y-intercept 'c' of the perpendicular line can be found by substituting into y = (-1/2)x + c.
The correct equation that satisfies these conditions is y = -1/2x + 7, none of the given options match this exact form, so the answer would be 'none of the above' if this were a forced choice. However, the original question does not provide the correct options given the calculations.
The correct option is not given.