Find the distance

Question

Line l contains the point(2,2) and (6,4) . Find the distance between the line and the point (1,9) . Leave your answer in radical form.

Answer 1

Slope of given line: (4-2) / (6-2) = 1/2

Equation of line: y - 2 = (1/2)(x - 2) So, y = (1/2)x + 1

Any line perpendicular to the given line has slope -1/(1/2) = -2.

Equation of line with slope -2 and containing (1,9): y - 9 = -2(x - 1). So, y = -2x + 11.

Next, find the point of intersection. P, of the two lines above.

Then, the distance from the given line to the point (1,9) is the distance from (1,9) to P.

Answer 2

To find the intersection of y=(1/2)x+1 and y=-2x+11, set

(1/2)x+1 = -2x+11 (multiply both sides by 2 to remove the fraction)

x+2 = -4x+22

Add 4x-2 to both sides:

5x = 20

DIvide both sides by 5:

x = 4

Since y = (1/2)x+1 = 0.5×4+1 = 2+1 = 3.

Checking: y=-2x+11=-2×4+11 = -8+11 = 3.

Using the distance formula,

√[(1-4)² + (9-3)²] = √[ 9 + 36 ] = √(45) = √(9*5) = 3√5.