The shortest distance between a point and a line is the length of the line segment that is perpendicular to the given line. We will need to find the perpendicular line and then we can find the intersection point. Finally, we can calculate the distance from the given point to the point of intersection.
Finding the Perpendicular Line
Perpendicular lines have negative reciprocal slopes. This means the product of their slopes is equal to -1.
m_1 * m_2=- 1
To help us identify the slope of the given line, let's first convert it into slope-intercept form.
The given line has a slope of 12.
y= 1/2x+7
By substituting this value into the equation above for m_1, we can find the slope of the perpendicular line.
We can add this value of b, along with the known slope, into the slope-intercept form to have a complete equation for the perpendicular line.
y= - 2x+ 9/2
Finding the Point of Intersection
To find the distance, we also need to know where the given line and the perpendicular line intersect. By setting up a system of equations, we can find the point of intersection. Note that we will use the slope-intercept form of the given equation.
y= 12x+7 y=- 2x+ 92
Since both equations have y isolated, it is most convenient to use the Substitution Method.
