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Find the shortest distance from the point (3, 3) to the line defined by the equation y = 14x - 2, while ensuring that the shortest distance is perpendicular to the given line."

User Weiyan
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Final answer:

To find the shortest distance from a point to a line while ensuring it is perpendicular to the line, you can use the formula for the distance between a point and a line. First, find the equation of the perpendicular line by using the point-slope form, then find the intersection point of the two lines. Finally, calculate the distance between the given point and the intersection point using the distance formula.

Step-by-step explanation:

To find the shortest distance from a point to a line while ensuring it is perpendicular to the line, you can use the formula for the distance between a point and a line. The given line is y = 14x - 2, so the slope of the line is 14. The line perpendicular to it will have a slope of -1/14, which is the negative reciprocal of 14. To find the equation of the perpendicular line, you can use the point-slope form y - y1 = m(x - x1), where (x1, y1) is the point (3, 3). Substitute the values into the equation and simplify to get y = -1/14x + 133/14.

Now, find the intersection point of the given line and the perpendicular line by setting the two equations equal to each other: 14x - 2 = -1/14x + 133/14. Solve for x and substitute the value of x back into either equation to find the corresponding y-coordinate. The intersection point is the closest point on the line to the given point.

Finally, calculate the distance between the given point and the intersection point using the distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2).

User Desiree
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