Finding distance from a point to a line

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Finding distance from a point to a line
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Choosing the closest point

• Which point on the line y0 is closest to point
  D?
• L (3.6,0) M (4,0) or N (4.25,0)
• Use the distance formula!!!!!
• DL ?
• DM ?
• DN ?

3

• The shortest distance from a point to a line, is
  the length of the perpendicular segment from the
  point to the line.

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Theorem 42-1

• Through a line and a point not on the line, there
  exists exactly 1 perpendicular line to the given
  line

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Theorem 42-2

• The perpendicular segment from a point to a line
  is the shortest segment from the point to the line

6

• The length of a perpendicular segment from a
  point to a line is referred to as the distance
  from a point to a line

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• What happens when you use the distance formula to
  find the distance between a point and a vertical
  line?
• The y coordinates are the same, so their
  difference is 0, so you only have to take the
  square root of the (x2-x1)2

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Finding the closest point on a line to a point

• Given the equation y 2x 1 and the point
  S(3,2), find the point on the line that is
  closest to S. Find the shortest distance from S
  to the line.
• 1. draw the line and the point on a graph
• 2. find the slope of the line m 2
• 3. find the slope of the line perpendicular to
  the given line m -1/2
• 4. use the slope to find more points on the
  perpendicular line
• 5.draw the line- the lines intersect at (1,3)
• 6. use the distance formula to find the distance
  between (3,2) and (1, 3)

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Theorem 42-3

• The perpendicular segment from a point to a plane
  is the shortest segment from the point to the
  plane.

10

• Because parallel lines are always the same
  distance from one another, the distance from any
  point on a line to a line that is parallel is the
  same, regardless of which point you pick

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Theorem 42-4

• If 2 lines are parallel, then all points on 1
  line are equidistant from the other line

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Find the shortest distance from yx2 and point
R(4,4)

• Follow steps
• Graph y x 2 and point R
• Find perp. slope
• From point R, find intersection point by using
  slope (3,5)
• find distance from point R (4,4) to (3,5)