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Parallel and Perpendicular

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To discover how slopes of parallel and perpendicular lines are related To learn the definitions of various quadrilaterals defined by their parallel and perpendicular ... – PowerPoint PPT presentation

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Title: Parallel and Perpendicular


1
Parallel and Perpendicular
  • To discover how slopes of parallel and
    perpendicular lines are related
  • To learn the definitions of various
    quadrilaterals defined by their parallel and
    perpendicular sides
  • To learn the definitions of inductive and
    deductive reasoning

2
Parallel lines are lines in the same plane
that never intersect. They are always the same
distance apart.
Perpendicular lines are lines that meet at a
right angle, that is, at an angle that measures
90.
3
Slopes
  • A rectangle has two pairs of parallel line
    segments and four right angles. When you draw a
    rectangle on the coordinate plane and notice the
    slopes of its sides, you will discover how the
    slopes of parallel and perpendicular lines are
    related.
  • Draw coordinate axes centered on graph paper.
    Each member of your group should choose one of
    the following sets of points. Plot the points and
    connect them, in order, to form a closed polygon.
    You should have formed a rectangle.
  • a. A(6, 20), B(13, 11), C(-5, -3), D(-12, 6)
  • b. A(3, 1), B(3, 7), C(9, 16), D(15, 8)
  • c. A(11, 21), B(17, 11), C(12, 3), D(16, 7)
  • d. A(3, 10), B(5, 22), C(7, 25), D(15, 7)

4
  • The slope of a line segment is the same as the
    slope of the line containing the segment.
  • What conjecture can you make about the slopes of
    parallel lines based on your answers to the
    previous two steps?

5
  • To find the reciprocal of a number, you write the
    number as a fraction and then invert it.
  • For example, the reciprocal of 2/3 is 3/2.
  • The product of reciprocals is 1.

6
  • Express the slope values of AB and BC as reduced
    fractions.
  • Express the slope values of AD and DC as reduced
    fractions.
  • What conjecture can you make about the slopes of
    perpendicular lines? What is their product? Check
    your conjecture by finding the slopes of any
    other pair of perpendicular sides in your
    rectangle.
  • On the coordinate plane, draw two new pairs of
    parallel lines that have the slope relationship
    you discovered in in this investigation. What
    figure is formed where the two pairs of lines
    intersect?

7
In the investigation, you made conjectures based
on studying examples. When you do this, you are
using inductive reasoning.
8
Example A
  • Triangle ABC (written as ?ABC) is formed by
    connecting the points (1, 3), (9, 5), and (10,
    1). Is it a right triangle?

9
In Example A you used the fact that perpendicular
lines have opposite reciprocal slopes to
determine that ABC is a right triangle. The
process of showing that certain statements or
conclusions follow logically from an initial
assumption or fact is called deductive reasoning.
10
Example B
  • Classify as specifically as possible the polygon
    formed by the pointsA(-4, 1), B(-2, 4), C(4,
    0), and D(-1, -1).

11
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