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.

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.

4. Solve the expression. Round your final answer to the nearest hundredths.
( 5

3 )

( 30

25 )

B(6,4)
A(2,1)
AB
C(6,1)

Find the lengths of the legs of ABC.
BC
 y

2 y
1
BC
BC
CA



5

1
4 x

2 x
1
CA

CA

4

2
2

Use the Pythagorean theorem to find the length of the hypotenuse AB.
     
 
2 
   
2
2
AB
 
 
2
2


16
25

9
AB

25
AB

5

4.
(5
-
3)
+
(30
-
25)
2
+
5
7
2. 65

The steps used in the warm up can be used to develop a general formula for the distance between two points.
A ( x
1
, y
1
)
B ( x
2
, y
2
)

A ( x
1
, y
1
)
B ( x
2
, y
2
)
C
A ( x
1
, y
1
)
B ( x
2
, y
2
)
What are the coordinates of C?
C ( x
2
, y
1
)

A ( x
1
, y
1
)
B ( x
2
, y
2
)
C ( x
2
, y
1
)
A ( x
1
, y
1
)
B ( x
2
, y
2
)
What are the coordinates of C?
C ( x
2
, y
1
)

A ( x
1
, y
1
)
AC
 x
2
B ( x
2
BC

, y
2
) y
2


C ( x
2 x
1
, y
1
)
What are the lengths of the triangle ’ s y
1 sides?
AC
 x
2
 x
1
BC
 y
2
 y
1

A ( x
1
, y
1
)
AC
AB
 x
2
B ( x
2
BC

,

C ( x
2 x
1
, y
2 y
2 y
1
)
)

What is the length of the triangle ’ s y
1 hypotenuse?
     
2
 
 
2



 x
2 x
2

 x
1 x
1
 
2  y y
2
2

 y
1 y
1


2
2
The length of the hypotenuse is equal to d the distance between points A and B.

 x
2
 x
1
  y
2
 y
1

2

creating a triangle and using the the hypotenuse. The hypotenuse of the between the two points.


Find the distance between (1,4) and (-
2,3).
d
Solution: d


( x
2
(

2
 x
1
)
2

1 )
2

 y
2
 y
1

2

 
2
Write the distance formula
Substitute d d

10

3 .
16
Simplify
Use a calculator


Find the distance between
1
2
,
1
4
 and
 
.

d
Solution:

( x
2
 x
1
)
2 
 y
2
 y
1

2 d



1
2 

 2
 

1
1
4 

 2 d




3
2 

 2




3
4 

 2 d d




45
16 



1 .
67
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) d3 d2
(3,2) d1
(2,0)

Example 3 Checking A Right
Triangle d 1

Solution:
Use the distance formula to find the lengths of the three sides.
( 3

2 )
2 
( 2

0 )
2

1

4

5 d 2

( 3

( 1 ))
2 
( 2

4 )
2

16

4

20 d 3

( 2

( 1 ))
2 
( 0

4 )
2

9

16

25

Example 3 Checking A Right
Triangle
(-1,4)
   
2 
   
2


5

20
25 d2
(3,2) d3

Next we find the sum of the squares of the lengths of the two shorter sides.

Example 3 Checking A Right
Triangle
(-1,4)
   
2 
   
2


5

20
25 d2
(3,2)


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, d3  
2
Thus, the given points are vertices of a right triangles.

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
(1,2)
(4,6)
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
(1,2)
(4,6)
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
(1,2)
(4,6) radius
=
5 radius=distance(d) d
 d

( x
2
 x
1
)
2 
 y
2
 y
1

2
( 4

1 )
2 
 
2 d
 d

( 3 )
2 
 
2
9

16 d
 d

5
25

Solve the following individually:
1.
2.
Find the distance between(-3,4) and
(5,4).
Find the distance between the two points:
1
3
,
1
6


,
2
3
,
8
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 d


( x
2
 x
1
)
2
( 5

(

3 ))
2

 y
2
 y
1

2

 
2 d

( 8 )
2 d

8

Solutions to individual problem #2 d

2. Find the distance between the two points: d



 3
3


 2




15
6


 2
( x
2
 x
1
)
2 
 y
2
 y
1

2 d
 d





2
3

1
3 

 2



8
3
1
6 

 2


 
2

3
1


 2



6
1


 2 d
 d
 d

1

25
4
4
4

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 d


( x
2
( 5

 x
1
3 )
2
)
2



 y
2
 y
1

2
4

2

2 d

( 2 )
2 
 
2 d

4

4 d

8 d d


2
4

2
2 r

2 2
(3,2)
(5,4)

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.

D(0,4)
Solution to trapezoid problem #1
C(2,8) B(5,9)
Draw the polygon whose vertices are
A(1,1),B(5,9),C(2,8), and D(0,4).
A(1,1)

D(0,4)
Slope Review
Solution to trapezoid problem #2
C(2,8) B(5,9)
Show that the polygon is a trapezoid by showing that only two of the sides are parallel.
A(1,1)

D(0,4)
Solution to trapezoid problem #2
C(2,8) B(5,9)
Show that the polygon is a trapezoid by showing that only two of the sides are parallel.
A(1,1)

D(0,4)
Solution to trapezoid problem #2
C(2,8)
A(1,1)
B(5,9)



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

m

2
D(0,4)
Solution to trapezoid problem #2
C(2,8)
A(1,1) m

2
B(5,9)
Slope of AB: m

9
5

1
1 m

2
Slope of CD: m

8
2


4
0 m

2
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.

D(0,4)
Solution to trapezoid problem #3
C(2,8)
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.

D(0,4)
Solution to trapezoid problem #3
C(2,8)
A(1,1)
B(5,9)
CB

CB

CB

CB

DA

DA

DA
DA


( 2

5 )
2
(

3 )
2 



1
8
2

9

2
9

1
10
( 0

1 )
2 

4

1

2
(

1 )
2 
 
2
1

9
10
CB

DA

10
Since CB and DA are equidistant, the trapezoid ABCD is isosceles.





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.


4
5

5
Add the following:
4


5
5
25

9
9
25
20
25

9
25
29
25


Reduce the expression
25
4

5
12
25
12
3
2
5
3
Back to problem




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.
A
Back to Lesson
D
B
C

slope
 slope
 rise y run

2 x
2
 y x
1
1

(1,1)
(7,9)
The slope is positive. Note that when a line has a positive slope it rises up left to right.

Find the slope of the line that passes through the points
(7,9) and (1,1).
slope slope slope



9
7
8
6 y x
2
2



1
1 y x
1
1 slope

4
3

Back to Lesson
(1,5)
(7,2)
The slope is negative. Note that when a line has a negative slope it falls left to right.

Find the slope of the line that passes through the points
(1,5) and (7,2).
slope slope slope




5
1
3 y x
6
2
2



2
7 y x
1
1 slope
 
1
2

Back to Lesson
D
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
A and C are parallel; they have the same slope.
If B and D have the same length, then the trapezoid is isosceles.
C