Final answer:
The distance from point P (3, 10) to line I is calculated using the slope of line I derived from given points and the perpendicular distance formula. The slope is found to be 1/2, leading to a line equation of y = (1/2)x + 1. The calculated distance is approximately 9.39 units, which is closest to option B (9.22 units), though this may involve a discrepancy.
Step-by-step explanation:
The question involves finding the distance from a point to a line in two-dimensional space, which is a topic in high school level mathematics dealing with geometry and analytical methods. To calculate the distance from point P (3, 10) to line I that passes through points (4, 3) and (-2, 0), one must first determine the equation of line I using the two given points to find the slope and y-intercept. Once the equation of the line is established, the shortest distance from point P to this line can be found using the perpendicular distance formula.
Let's follow the steps to determine this distance. First, we calculate the slope of line I using the points (4, 3) and (-2, 0):
- Slope (m) = (y2 - y1) / (x2 - x1)
- m = (3 - 0) / (4 - (-2))
- m = 3 / 6
- m = 1/2
Then the equation of line I, in slope-intercept form (y = mx + b), can be derived using point (4, 3) and the slope:
- 3 = (1/2)(4) + b
- b = 3 - 2
- b = 1
- So, the equation of line I is y = (1/2)x + 1
Now, the formula for the distance from a point to a line is:
- Distance = |Ax + By + C| / sqrt(A^2 + B^2)
- Here, A = -m, B = 1, and C = -b from the line equation's standard form Ax + By + C = 0
For our line, y = (1/2)x + 1, in standard form is:
- -x/2 + y - 1 = 0
- A = -(-1/2), B = 1, C = -1
- Distance = |(1/2)(3) + 1(10) - 1| / sqrt((1/2)^2 + 1^2)
- Distance = |1.5 + 10 - 1| / sqrt(0.25 + 1)
- Distance = |10.5| / sqrt(1.25)
- Distance = 10.5 / 1.118
- Distance = 9.39 (rounded to two decimal places)
The closest answer choice to our calculated distance is 9.22 units (Option B), but note that this calculation does not match any of the provided answer choices exactly, which may suggest an error in the provided options or a calculation mistake.