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Understanding Perpendicular Distances in Geometry

This section covers the concept of determining the distance between a point and a line, as well as between two parallel lines. It introduces Postulate 3.6, which states that there is exactly one line through a given point that is perpendicular to a given line. The process involves finding the equation of the line, writing the perpendicular line's equation, and determining their intersection point to calculate the distance. Examples illustrate how to apply these principles, solidifying understanding of geometric relationships.

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Understanding Perpendicular Distances in Geometry

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Presentation Transcript

  1. Geometry 3.6 Perpendiculars and Distance

  2. The distance between a line and a point not on the line is the length of the segment perpendicular to the line from the point.

  3. Postulate 3.6Perpendicular Postulate • If given a line and a point not on the line, then there exists exactly one line through the point that is perpendicular to the given line.

  4. Ex. 1 • Copy the figure and construct the segment that represents the distance from Q to segment RS. S R Q P

  5. Steps to find distance if given a line and a point • 1) Find the equation for the given line. • 2) Write an equation for line perpendicular to the given line through the given point. • 3) Find the point of intersection between the two lines. • 4) Use the distance formula to find the distance between the given point and intersection.

  6. Example 2 • Line s contains points (0,0) and (-5,5). Find the distance between line s and point V(1,5)

  7. The distance between two parallel lines is the perpendicular distance between one of the lines and any point on the other line. • Theorem 3.9: In a plane, if two are lines are equidistant from a third line, then the two lines are parallel to each other.

  8. Steps to find distance between two parallel lines. • 1. Write an equation for a line perpendicular to the parallel lines. Use the same y-intercept as a given line. • 2. Solve a system of equation to find the perpendicular line intersects the other line. • 3. Use the distance formula to find the distance between the points of intersection.

  9. Example 3 • Find the distance between the parallel lines a and b with equations y= 2x +3 and y =2x -1

  10. Homework • Page 218: 9-27 odds

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