Transcript A(2,1)

Michael Reyes
MTED 301 Section 1-2.
Subject: Geometry
Grade Level:9-10
Lesson:
The Distance Formula
Objective:
California Mathematics Content Standard
Geometry 17.0:
Students prove theorems by using coordinate geometry, including the
midpoint of a line segment, the distance formula, and various forms of
equations of lines and circles.
California Common Core Standard
Geometry Congruence 9.0:
Prove theorems about lines and angles.
Materials: Larson, R., Boswell, L., Kanold, T., Stiff, L. McDougall
Littell.(2001). Algebra I, pp. 745-751.
Warm-Up
1.
2.
3.
Plot and label A(2,1) and B(6,5) and
C(6,1) on graph paper. Connect the
points to form a right triangle with AB
as the hypotenuse.
Find the lengths of the legs of triangle
ABC. This means find the lengths of BC
and CA.
Use the Pythagorean theorem to find
the length of the hypotenuse AB.
Warm-Up cont.
4. Solve the expression. Round your final
answer to the nearest hundredths.
( 5  3 )  ( 30  25 )
Warm-Up Solution to #1
B(6,4)
AB
A(2,1)
C(6,1)
Warm-Up Solution to #2
Find the lengths of the legs of
BC  y 2  y 1
BC  5  1
BC  4
CA  x 2  x1
CA  4  2
CA  2
ABC.
Warm-Up Solution to #3
Use the Pythagorean theorem to find the
length of the hypotenuse AB.
AB   BC   CA 
 AB   4   3 
AB   16  9
AB   25
2
2
2
2
2
2
AB 
AB  5
25
2
2
Warm-Up Solution to #4
4.
(5- 3)+ (30 - 25)
2+5
7
2. 65
The Distance Formula
The steps used in the warm up can be
used to develop a general formula for the
distance between two points.
A ( x1 , y 1 )
B ( x2 , y2 )
The Distance Formula
B ( x2 , y2 )
A ( x1 , y 1 )
C
A ( x1 , y 1 )
B ( x2 , y2 )
What are the
coordinates of C?
C ( x 2 , y1 )
The Distance Formula
B ( x2 , y2 )
A ( x1 , y 1 )
C ( x 2 , y1 )
A ( x1 , y 1 )
B ( x2 , y2 )
What are the
coordinates of C?
C ( x 2 , y1 )
The Distance Formula
B ( x2 , y2 )
What are the lengths
of the triangle’s
sides?
BC  y 2  y1
AC

x

x
2
1
A ( x1 , y 1 )
C ( x 2 , y1 )
BC  y 2  y1
AC  x 2  x1
The Distance Formula
What is the length of
the triangle’s
hypotenuse?
B ( x2 , y2 )
AB
BC  y 2  y1
A ( x1 , y 1 )
C ( x 2 , y 1 )  AB 2   AC 2  BC
AC  x 2  x1
2
AB 

2
  x 2  x1    y 2  y1 
2
 AB  
 x 2  x1 
2
  y 2  y1 
The length of the hypotenuse is equal to
the distance between points A and B.
d 
 x 2  x1 
2
  y 2  y1 
2
2
2
Vocabulary Check
Distance Formula
The _____________can
be obtained by
creating a triangle and using the
Pythagorean Theorem
________________to
find the length of
the hypotenuse. The hypotenuse of the
Distance
triangle will be the ___________
between the two points.
Example 1

Find the distance between (1,4) and (2,3).
Solution:
d  ( x 2  x1 )   y 2  y1  Write the distance formula
2
2
d  (  2  1)  3  4  Substitute
2
2
d  10
Simplify
d  3 . 16
Use a calculator
Example 2

Find the distance between
1 1
 ,  and  2 ,1 .
2 4
Example 2
Solution:
d  ( x 2  x1 )   y 2  y 1 
2
2
2
2
1 

d   2    1 
2 

2
3 3
d     
2 4
 45 
d  
 16 
d  1 . 67
2
1

4
Write the distance formula
Substitute
Simplify
Simplify
Use a calculator
Example 3 Checking A Right
Triangle

Decide whether the points (3,2),(2,0), and
(-1,4) are vertices of a right triangle.
Begin by graphing the triangle with the
given vertices.
Example 3 Checking A Right
Triangle
Does this look like a right triangle? We can apply the
distance formula to check if it is truly a right triangle.
(-1,4)
d2
d3
(3,2)
d1
(2,0)
Example 3 Checking A Right
Triangle
Solution:
Use the distance formula to find the
lengths of the three sides.
1 4

16  4 
20
9  16
25
(3  2 )  ( 2  0 )
d2 
( 3  (1))  ( 2  4 )
d3 
2
2
( 2  (1))  ( 0  4 ) 
2
2
2
2


d1 

5
Example 3 Checking A Right
Triangle
d 1
2
 d 2  
2
 5  
2
 5  20
 25
(-1,4)
d2
d3
(3,2)
d1
(2,0)

2
20

Next we find the
sum of the squares of
the lengths of the
two shorter sides.
Example 3 Checking A Right
Triangle
d 1
2
 d 2  
2
 5  
2
 5  20
 25
(-1,4)
d2
d3
(3,2)
d1
(2,0)

2
20
The sum of the
squares of the
lengths of the
shorter sides is 25.
 This is equal to the
square of the length
of the longest side,


 25  
Thus, the given points are vertices of a right
triangles.
2
Example 4 Application of the Distance
Formula
How can you use the
distance formula to
solve problems like
the following one:
The point (1,2) lies
on a circle. What is
the length of the
radius of this circle if
the center is located
at (4,6)?
Circles Review
Example 4 Application of the Distance
Formula
(4,6)
(1,2)
How can you use the
distance formula to
solve problems like
the following one:
The point (1,2) lies
on a circle. What is
the length of the
radius of this circle if
the center is located
at (4,6)?
Example 4 Application of the Distance
Formula
(4,6)
(1,2)
The point (1,2) lies on
a circle. What is the
length of the radius of
this circle if the center
is located at (4,6)?
The length is equal to
the distance between
the center point and
any point located on
the edge of the circle.
Example 4 Application of the Distance
Formula
radius=distance(d)
d  ( x 2  x1 )   y 2  y 1 
2
(4,6)
radius = 5
(1,2)
d  ( 4  1)   6  2 
2
d  ( 3 )  4 
2
d 
9  16
d 
25
d 5
2
2
2
Solve the following individually:
1.
2.
Find the distance between(-3,4) and
(5,4).
Find the distance between the two
points:  1 , 1  ,   2 , 8 .
3 6 
3.
3 3
The point (5,4) lies on a circle. What is
the length of the radius of this circle if
the center is located at (3,2)?
Solutions to individual problem #1
Find the distance between(-3,4) and
(5,4).
Solution:
d  ( x 2  x1 )   y 2  y 1 
2
d  ( 5  (  3 ))   4  4 
2
d  (8 )
d 8
2
2
2
Solutions to individual problem #2
2. Find the distance between the two points:
d  ( x 2  x1 )   y 2  y 1 
2
2
2
 2 1 8 1
d        
 3 3 3 6
2
  2  1   16  1 
d  
 

 3   6 
2
2

d  

3   15 
  
3  6 
d  1
25
2
4
d 
4

4
2
d 
25
4
29
4
Help with Fractions
Solutions to individual problem #3

The point (5,4) lies
on a circle. What is
the length of the
radius of this circle if
the center is located
at (3,2)?
(5,4)
(3,2)
Solutions to individual problem #3
Distance(d)=length of radius
d 
( x 2  x1 )   y 2  y 1 
d 
( 5  3 )  4  2 
d 
( 2 )  2 
d 
44
d 
8
d 
2
2
2
4
d  2 2
2
2
2
(5,4)
r  2 2
(3,2)
2
Trapezoid Review
Solve the following problem with a
partner:
1.
2.
3.
Draw the polygon whose vertices are
A(1,1),B(5,9),C(2,8), and D(0,4).
Show that the polygon is a trapezoid by
showing that only two of the sides are
parallel.
Use the distance formula to show that
the trapezoid is isosceles.
Solution to trapezoid problem #1
C(2,8)
D(0,4)
A(1,1)
B(5,9)
Draw the polygon
whose vertices are
A(1,1),B(5,9),C(2,8),
and D(0,4).
Slope Review
Solution to trapezoid problem #2
C(2,8)
D(0,4)
A(1,1)
B(5,9)
Show that the polygon
is a trapezoid by
showing that only two
of the sides are
parallel.
Solution to trapezoid problem #2
C(2,8)
D(0,4)
A(1,1)
B(5,9)
Show that the polygon
is a trapezoid by
showing that only two
of the sides are
parallel.
Solution to trapezoid problem #2
Two side are parallel if
they have the same
slope.
 CB has a positive slope
and DA has a negative
slope.
 The slopes that are left
to check are those of
AB and CD

C(2,8)
D(0,4)
A(1,1)
B(5,9)
Solution to trapezoid problem #2
Slope of AB:
C(2,8)
B(5,9)
m 2
D(0,4)
m 2
m 
9 1
5 1
m 2
Slope of CD:
m 
84
20
m 2
A(1,1)
Since the slopes of AB and CD are
equal, then the two sides are
parallel. The polygon is a trapezoid
by definition of a trapezoid.
Solution to trapezoid problem #3
C(2,8)
D(0,4)
A(1,1)
B(5,9)
Use the distance
formula to show that
the trapezoid is
isosceles.
We must show that
CB and DA have the
same distance to
demonstrate that
trapezoid ABCD is
isosceles.
Solution to trapezoid problem #3
C(2,8)
D(0,4)
A(1,1)
B(5,9)
CB 
( 2  5 )  8  9 
CB 
(  3 )   1
CB 
9 1
CB 
10
DA 
( 0  1)   4  1 
DA 
(  1)   3 
DA 
1 9
DA 
10
CB  DA 
2
2
2
2
2
2
2
2
10
Since CB and DA are equidistant, the trapezoid ABCD is
isosceles.
What we learned today:
How the distance formula is derived.
 How to find the distance between to
points.
 How to check if a triangle is a right
triangle using the distance formula
 How to prove properties of shapes using
the distance formula.

Fill in the blanks and turn in before you
leave.
The distance formula can be obtained by
creating a triangle and using the
________________to find the length of
the hypotenuse. The hypotenuse of the
triangle will be the ___________
between the two points.
The distance between the center of a
circle and any point on the circle is the
______ of the circle.
The End
Help with fractions

Add the following:
4

5
25
45
55
20
25
29
25
9

9

25
9
25
Help with Fractions

Reduce the expression
12
25
12
25
4
3
5
2 3
5
Back to problem
Back to Lesson
Circles Review
What is the definition
of a circle?
 A circle is the set of all
points equidistant
from a center point.
 AD, BD, and CD are all
equidistant and radii of
the circle.

B
D
A
C
Slope Review
slope 
slope 
rise
run
y 2  y1
x 2  x1
Slope Review

(7,9)
Find the slope of the
line that passes
through the points
(7,9) and (1,1).
y 2  y1
slope 
x 2  x1
slope 
(1,1)
The slope is positive. Note that when a
line has a positive slope it rises up left to
right.
slope 
slope 
9 1
7 1
8
6
4
3
Back to Lesson
Slope Review

(1,5)
(7,2)
Find the slope of the
line that passes
through the points
(1,5) and (7,2).
y 2  y1
slope 
x 2  x1
slope 
slope 
The slope is negative. Note that when a
line has a negative slope it falls left to
right.
52
1 7
3
6
1
slope  
2
Back to Lesson
Trapezoid Review
A trapezoid is a quadrilateral with two
sides parallel.
If both legs are the same length, this is
called an isosceles trapezoid, and both
base angles are the same.
A
B
D
A and C are parallel; they
have the same slope.
If B and D have the same
length, then the trapezoid
is isosceles.
C