Answer: ²/₅ √(245) ≡ 6.26
Step-by-step explanation:
The shortest distance from a point to a line is along the perpendicular line that joins the point to that line. Therefore, if we can find that perpendicular line, then we can calculate the distance from the point, to the line.
Find the slope of the perpendicular line
When two lines are perpendicular, the product of their slopes is -1. This means that the slopes are negative-reciprocals of each other.
⇒ since the slope of y = -2x + 1 is -2
then the slope of the perpendicular line (m) = ¹/₂
Determine the equation of the perpendicular line
We can now use the point-slope form (y - y₁) = m(x - x₁)) to write the equation for this line:
⇒ y - 3 = ¹/₂ (x - 6)
Determine the point of intersection between the two lines
If we solve the equations simultaneously, we can find the point where the two lines intersect (the point of closest contact):
y - 3 = ¹/₂ (x - 6) [perpendicular line]
y = -2x + 1 [original line]
(-2x + 1) - 3 = ¹/₂ (x - 6) [by substituting og line into the perpen line]
-2x - 2 = ¹/₂ x - 3
-⁵/₂ x = - 1
x = ²/₅
when x = ²/₅, y = –2 ( ²/₅) + 1
= ¹/₅
∴ the lines intersect at the point (²/₅, ¹/₅).
Find the distance between the two points
The distance between two points = √[(x₂ - x₁)² + (y₂ - y₁)²]
⇒ the distance between (6, 3) and (²/₅, ¹/₅):
= √[(6 - ²/₅)² + (3 - ¹/₅)²]
= ²/₅ √(245) ≡ 6.26
∴ the distance from the point (6, 3) to the line y = –2x + 1 is ²/₅ √(245) or ≈ 6.26.
To test parts of my answer, I have included a Desmos Graph that I graphed using the information provided in the question and my answer.
