Final answer:
The equation of the line perpendicular to the line passing through the points (9,5) and (10,7) and containing the point (4,6) is y = -1/2x + 8.
Step-by-step explanation:
To find the equation of the line perpendicular to the line passing through the points (9,5) and (10,7), we first need to find the slope of the original line. The slope of a line passing through two points, (x1, y1) and (x2, y2), is given by the formula m = (y2 - y1) / (x2 - x1). Substituting the coordinates of the given points, we find that the slope of the original line is m = (7 - 5) / (10 - 9) = 2.
Since the line we want to find is perpendicular to the original line, its slope is the negative reciprocal of the original line's slope, which is -1/2. Now that we have the slope of the line we want to find and a point it passes through, (4,6), we can use the point-slope form of a linear equation, y - y1 = m(x - x1), to find the equation of the line. Substituting the values, we get y - 6 = (-1/2)(x - 4). Simplifying the equation, we find y - 6 = -1/2x + 2. Rearranging the terms, we get the equation of the line as y = -1/2x + 8.