3.6 Prove Theorems About Perpendicular Lines Objectives: Assignment: Challenge Problems

3.6 Prove Theorems About Perpendicular Lines
Objectives:
1. To find the distance
from a point to a line
2. To construct
perpendicular lines
with a compass and
straightedge
3. To prove theorems
about perpendicular
lines
Assignment:
• Challenge Problems
Objective 1
You will be able to find the
distance from a point to a line
Warm-Up, 1
What is the
distance from
point P to line l?
Warm-Up, 2
What is the
shortest
distance from P
to line l?
Distance from a Point to a Line
The distance from a point to a line is the
length of the perpendicular segment from
the point to the line. This is the shortest
distance from the point to the line.
Distance Between Parallel Lines
Likewise, the distance between two
parallel lines is the length of any
perpendicular segment joining the two
lines.
Objective 2
You will be able to construct perpendicular
lines with a compass and straightedge
Perpendicular Postulate
If we wanted to measure
the distance between a
point and a line, we can
employ the
Perpendicular
Postulate, a compass,
and a straightedge,
since there exists only
one perpendicular line
from a point to a line.
Perpendicular: Point Not on Line
1. Draw a line and
point P not on the
line.
Perpendicular: Point Not on Line
2. Using P as center,
draw an arc that
intersects the line
twice. Label the
points of
intersection A and
B.
Perpendicular: Point Not on Line
3. Using the same
compass setting,
draw two
intersecting arcs
below the line,
one centered at
A, the other
centered at B.
Label the point of
intersection Q.
Perpendicular: Point Not on Line
4. Draw a line
through points P
and Q.
While We’re At It…
The previous
construction assumed
that you were drawing
a perpendicular
through a point not on
a line. What if the
point was actually on
the line? We can still
use our compass and
straightedge to draw
the perpendicular line.
Perpendicular: Point On Line
1. Draw a line and a
point P on the line.
Perpendicular: Point On Line
2. Using P as center,
draw an arc below
the line that
intersects the line
twice. Label the
points of intersection
A and B.
Perpendicular: Point On Line
3. Using the same
compass setting
(but larger than the
first one), draw two
intersecting arcs
below the line, one
centered at A, the
other centered at B.
Label the point of
intersection Q.
Perpendicular: Point On Line
4. Draw a line from
through P and Q.
Objective 1
You will be able to find the
distance from a point to a line
Example 1
Find the
distance from
(−4, 3) to the
line
1
y  x  2.
2
Distance from a Point to a Line
The distance from a point (x1, y1) to the line
Ax + By + C = 0 is given by the formula
6
d
Ax1  By1  C
A B
2
2
(x 1, y 1)
4
2
5
-2
Example 1 Revisited
Find the
distance from
(−4, 3) to the
line
1
y  x  2.
2
Find the square root of
. This number is
the denominator.
Evaluate the equation at
the given point and then
take the absolute value.
This number is the
numerator.
Put the equation into
general form:
.
Find the Distance
To find the distance from a point to a line:
Example 2
Find the distance from (4, −2) to the line
3
y   x  5.
4
Objective 1
You will be able to develop, prove,
and use theorems about
perpendicular lines
Theorems Galore!
If two lines intersect to form a linear pair of
congruent angles, then the lines are
perpendicular.
Proof Hints:
• Write an equation based
on the angles forming a
linear pair.
• Do some substitution and
solve for one of the
angles. It should be 90°.
Theorems Galore!
If two lines are perpendicular, then they
intersect to form four right angles.
Proof Hints:
• Use definition of
perpendicular lines to find
one right angle.
• Use vertical and linear pairs
of angles to find three more
right angles.
Theorems Galore!
If two sides of two adjacent acute angles are
perpendicular, then the angles are
complementary. Proof Hints:
• Use definition of perpendicular
to get the measure of ABC.
• Use Angle Addition Postulate,
Substitution, and Definition of
complementary angles to finish
the proof.
Theorems Galore!
Perpendicular Transversal Theorem
If a transversal is perpendicular to one of two
parallel lines, then it is perpendicular to the
Proof Hints:
other.
j
• Use definition of perpendicular
lines to find one right angle.
• Use Corresponding Angles
Postulate to find a right angle on
the other line.
Theorems Galore!
Lines Perpendicular to a Transversal Theorem
In a plane, if two lines are perpendicular to
the same line, then they are parallel to
Proof Hints:
each other.
• Use definition of perpendicular
lines to find a right angle on each
parallel line.
• Use Converse of Corresponding
Angles Postulate to prove the
lines are parallel.
Parallel Lines the Right Way
Finally, we can
use the Lines
Perpendicular
to a
Transversal
Theorem to
construct a
couple of
parallel lines.
Parallel Lines the Right Way
1. Construct a
line with
points A and
B.
Parallel Lines the Right Way
2. Using the
same
compass
setting
construct
two arcs, one
centered at
A and one
centered at
B.
Parallel Lines the Right Way
3. Construct
points of
intersection
C, D, E, and
F.
Parallel Lines the Right Way
4. Using a slightly
larger compass
setting,
construct two
sets of
intersecting
arcs: one set
centered at C
and D; the
other centered
at E and F.
Parallel Lines the Right Way
5. Construct
points of
intersection G
and H.
6. Draw a line
through A and
G.
7. Draw a line
through B and
H.
3.6 Prove Theorems About Perpendicular Lines
Objectives:
1. To find the distance
from a point to a line
2. To construct
perpendicular lines
with a compass and
straightedge
3. To prove theorems
about perpendicular
lines
Assignment:
• Challenge Problems