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} \cdot m_{2} = - 1

The given line has a slope of

1 .

y = x + 4 \Leftrightarrow y = 1 x + 4

By substituting this value into the equation above for

m_{1},

we can find the slope of the perpendicular line.

m_{1} \cdot m_{2} = - 1

Substitute

$m_{1} = 1$

1 \cdot m_{2} = - 1

OneMult

$1 \cdot a = a$

m_{2} = - 1

Now, to find the equation of the perpendicular line, we can substitute the known point in the slope-intercept form, using that the slope is

- 1 .

y = - 1 x + b

SubstituteII

$x = 6$ , $y = 4$

4 = - 1 (6) + b

▼

Solve for

b

MultNegPos

$(- a) b = - a b$

4 = - 6 + b

AddEqn

$LHS + 6 = RHS + 6$

10 = b

RearrangeEqn

Rearrange equation

b = 10

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 = - 1 x + 10 \Leftrightarrow y = - x + 10

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.

Exercise