Final answer:
To construct a line perpendicular to line m through point P, find the slope of line m and then find the negative reciprocal of that slope. The equation of the line perpendicular to m through P is y - y1 = m' * (x - x1), and the distance from P to m is 3 / sqrt(2).
Step-by-step explanation:
To construct a line perpendicular to m through point P, we need to find the slope of line m and then find the negative reciprocal of that slope.
First, let's find the slope of line m using the coordinates of two points on the line:
m = (y2 - y1) / (x2 - x1) = (0 - (-3)) / (3 - 0) = 3 / 3 = 1.
So the slope of line m is 1.
Now, to find the slope of the line perpendicular to m, we take the negative reciprocal of 1, which is -1.
The equation of the line perpendicular to m through point P is y - y1 = m' * (x - x1), where m' is the slope of the perpendicular line.
Plugging in the values for point P, we get y - 2 = -1 * (x - 1).
To find the distance from point P to line m, we can use the formula for the distance between a point and a line.
The formula is:
distance = |Ax + By + C| / sqrt(A^2 + B^2),
where A, B, and C are the coefficients of the equation of the line, and (x, y) is a point on the line. In this case, the equation of line m is y = mx, where m = 1.
So, A = -1, B = 1, and C = 0. Plugging in the values for point P, we get:
distance = |(-1)(1) + (1)(2) + (0)| / sqrt((-1)^2 + 1^2) = |1 + 2 + 0| / sqrt(1 + 1) = 3 / sqrt(2).
Therefore, the distance from point P to line m is 3 / sqrt(2).