Honors Geometry

1
Honors Geometry

• Lesson 1.5
• Coordinate Geometry
• and
• Noncoordinate Geometry

2
Coordinate Geometry and Noncoordinate Geometry

• Goal 1 How to use slope in coordinate geometry
• Goal 2 Explore relationships between points,
  segments, lines, and planes in noncoordinate
  geometry
• Plans for constructing any real-life object
  usually require an accurate drawing of the final
  product. Coordinate geometry and noncoordinate
  geometry can help you draw the plans.

3
Using Slope in Coordinate Geometry

• The real number line is a one-dimensional system
  in which points have a single coordinate.
• The coordinate plane is a two-dimensional system
  in which points have two coordinates.

4
Using Slope in Coordinate Geometry

• The slope, m, of the line that contains (x1,y1)
  and (x2,y2) is m
• Two nonvertical lines are parallel if and only if
  they have the same slope.
• Two nonvertical lines with slopes of m1 and m2
  are perpendicular if and only if m2 is the
  negative reciprocal of m1 (that is m1m2 -1)

5
Finding the Slope of a Line

• Line l contains A(2,3) and B(-1,4). Find the
  slope of line l.

6
Finding the Slope of a Line

• m

7
Parallel and Nonparallel Lines

• Decide whether the lines are parallel,
  perpendicular, or neither. Remember the slope of
  the line y mx b is equal to m.
• A. Line p y 2x 4 Line q y -½x 4
• B. Line p y ½x 2 Line q y ½x -2
• C. Line p y 2x 1 Line q y -2x -1

8
Parallel and Nonparallel Lines

• A. m 2 and m -½, Because -½ is the negative
  reciprocal of 2, it follows that the lines are
  perpendicular.
• B. m ½ and m ½ the slopes are the same so the
  lines are parallel.
• C. m 2 and m -2, the lines are neither
  parallel or perpendicular

9
Parallel and Nonparallel Lines
10
Relationships in Noncoordinate Geometry

• Coordinate geometry, (also called analytic
  geometry) uses coordinate systems to study the
  properties of segments, lines, planes, and other
  figures.
• Historically, much of geometry was developed
  without a coordinate system. This type of
  geometry is called Euclidean geometry or
  noncoordinate geometry after the Greek
  mathematician Euclid.

11
Lines in a Noncoordinate Plane

• Of the lines p, q, r, and s , which appear to be
  parallel? Which appear to be perpendicular?

12
Lines in a Noncoordinate Plane

• Lines p and q appear to be parallel
• Line r appears to be perpendicular to lines p and
  q
• Line s is neither parallel nor perpendicular to
  any of the other lines

13
Points, Lines and Planes in Space

• Ascending and Descending is by Dutch artist
  Maurits Escher(1898-1972). Escher is well known
  for using tricks of perspective and scale to
  create optical illusions in his art. In this
  drawing, the staircase appears to be always
  ascending (or always descending).

14
Points, Lines and Planes in Space

• Describe some of the apparent relationships
  between the point, lines, and planes labeled in
  the impossible staircase drawing at the left.

15
Points, Lines and Planes in Space

• The planes, or flat surfaces, labeled A,B, and C
  appear to be parallel.
• Point P and lines n and m appear to be coplanar.
  They all lie in plane E.
• Line m appears to be perpendicular to line l and
  to plane D.
• Plane D appears to be perpendicular to Plane E.

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The End