Final answer:
To find the distance between line l and point p, we can use the formula for the distance between a point and a line. We need to find the equation of line l using the two given points, then find the equation of the line perpendicular to l that passes through point p. The distance between line l and point p is equal to the distance between point p and the intersection of line l and the perpendicular line.
Step-by-step explanation:
To find the distance between line l and point p, we can use the formula for the distance between a point and a line. First, we need to find the equation of line l using the two given points. The slope of the line can be found by taking the difference in y-coordinates divided by the difference in x-coordinates. In this case, the slope is (4-2)/(5-1) = 2/4 = 1/2. Using the point-slope form of a linear equation, we can plug in one of the points and the slope to find the equation of the line: y - 2 = (1/2)(x - 1). To find the distance between the line and point p, we need to find the perpendicular distance from point p to line l. We can use the formula for the perpendicular distance between a point and a line.
First, we need to find the equation of the line perpendicular to l that passes through point p. The slope of this line is the negative reciprocal of the slope of l, which is -2. Using the point-slope form of a linear equation, we can plug in the coordinates of point p and the slope to find the equation of the perpendicular line: y - 7 = -2(x - 1).
To find the intersection point of line l and the perpendicular line, we can set the two equations equal to each other and solve for x and y. Solving the system of equations gives us the coordinates of the intersection point: x = -1/3 and y = 25/3.
Finally, we can find the distance between point p and the intersection point using the distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2). Plugging in the coordinates of point p and the intersection point, we get: d = sqrt((-1/3 - 1)^2 + (25/3 - 7)^2). Simplifying this expression gives us the distance between line l and point p.