Michael ReyesMTED 301 Section 1-2.Subject: GeometryGrade Level:9-10Lesson:The Distance FormulaObjective:California Mathematics Content StandardGeometry 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 StandardGeometry 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. 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.

2. Find the lengths of the legs of triangle ABC. This means find the lengths of BC and CA.

3. 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.

)2530()35(

Warm-Up Solution to #1

A(2,1) C(6,1

)

B(6,4)

AB

Warm-Up Solution to #2

Find the lengths of the legs of ABC.BC 12 yy

15 BC4BC

CA 12 xx

24 CA

2CA

Warm-Up Solution to #3

Use the Pythagorean theorem to find the length of the hypotenuse AB. 222

CABCAB

22234 AB

9162

AB

252AB

25AB

5AB

Warm-Up Solution to #4

4.

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.

),(

),(

22

11

yxB

yxA

The Distance Formula

What are the coordinates of C?

),(

),(

22

11

yxB

yxA

),( 11 yxA

),( 22 yxB

C

),( 12 yxC

The Distance Formula

What are the coordinates of C?

),(

),(

22

11

yxB

yxA

),( 11 yxA

),( 22 yxB

),( 12 yxC

),( 12 yxC

The Distance Formula

What are the lengths of the triangle’s sides?

),( 11 yxA

),( 22 yxB

12 xxAC

12 yyBC ),( 12 yxC

12 xxAC

12 yyBC

The Distance Formula

What is the length of the triangle’s hypotenuse?

),( 11 yxA

),( 22 yxB

),( 12 yxC

12 xxAC

12 yyBC 222

BCACAB

2122

12

2yyxxAB

2122

12 yyxxAB The length of the hypotenuse is equal to the distance between points A and B.

2122

12 yyxxd

AB

Vocabulary Check

The _____________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.

Distance Formula

Pythagorean Theorem

Distance

Example 1Find the distance between (1,4)

and (-2,3).Solution:

2122

12 )( yyxxd

22 43)12( d

10d16.3d

Write the distance formulaSubstitute

Simplify

Use a calculator

Example 2Find the distance between

.1,24

1,2

1and

Example 2

Solution: 212

212 )( yyxxd Write the distance

formula22

4

11

2

12

d Substitute

22

4

3

2

3

d

16

45d

67.1d

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

(-1,4)

(2,0)

(3,2)d2

d3 d1

Does this look like a right triangle? We can apply the distance formula to check if it is truly a right triangle.

Example 3 Checking A Right Triangle

Solution:Use the distance formula to find the lengths of the three sides.

22 )02()23(1 d

22 )40())1(2(3 d

22 )42())1(3(2 d 416

169

41 5

20

25

Example 3 Checking A Right Triangle

Next we find the sum of the squares of the lengths of the two shorter sides.

(-1,4)

(2,0)

(3,2)d2

d3 d1

2222 20521 dd

20525

Example 3 Checking A Right Triangle

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,

(-1,4)

(2,0)

(3,2)d2

d3 d1

2222 20521 dd

20525

225Thus, 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)?

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)?

(1,2)

(4,6)

Circles Review

Example 4 Application of the Distance Formula

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.

(1,2)

(4,6)

Example 4 Application of the Distance Formula

radius=distance(d)

(1,2)

(4,6)

2122

12 )( yyxxd

22 26)14( d

22 4)3( d

169d

25d5d

Solve the following individually:1. Find the distance between(-3,4)

and (5,4).2. Find the distance between the

two points:

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)?

.3

8,3

2,

6

1,3

1

Solutions to individual problem #1

Find the distance between(-3,4) and (5,4).

Solution:

2122

12 )( yyxxd

22 44))3(5( d2)8(d

8d

Solutions to individual problem #2

2122

12 )( yyxxd

22

6

1

3

8

3

1

3

2

d

22

6

116

3

12

d

22

6

15

3

3

d

4

251d

4

25

4

4d

4

29d

2. Find the distance between the two points:

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

(5,4)

(3,2)

Distance(d)=length of radius

2122

12 )( yyxxd

22 24)35( d

22 2)2( d

44d

8d

22d24 d

22r

Solve the following problem with a partner:1. Draw the polygon whose

vertices are A(1,1),B(5,9),C(2,8), and D(0,4).

2. Show that the polygon is a trapezoid by showing that only two of the sides are parallel.

3. Use the distance formula to show that the trapezoid is isosceles.

Trapezoid Review

Solution to trapezoid problem #1

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

A(1,1)

B(5,9)C(2,8)

D(0,4)

Solution to trapezoid problem #2

Show that the polygon is a trapezoid by showing that only two of the sides are parallel.

A(1,1)

B(5,9)C(2,8)

D(0,4)

Slope Review

Solution to trapezoid problem #2

Show that the polygon is a trapezoid by showing that only two of the sides are parallel.

A(1,1)

B(5,9)C(2,8)

D(0,4)

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

A(1,1)

B(5,9)C(2,8)

D(0,4)

Solution to trapezoid problem #2

Slope of AB:

Slope of CD:

A(1,1)

B(5,9)C(2,8)

D(0,4)

2m

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

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.

A(1,1)

B(5,9)C(2,8)

D(0,4)

Solution to trapezoid problem #3

A(1,1)

B(5,9)C(2,8)

D(0,4)

22 98)52( CB

22 1)3( CB

19CB

10CB

22 14)10( DA

22 3)1( DA

91DA

10DA10DACB

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 fractionsAdd the following:

25

9

5

4

25

9

55

54

25

9

25

20

25

29

Help with FractionsReduce the expression

25

12

25

12

5

34

5

32Back to problem

Circles ReviewWhat 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.

C

B

AD

Back to Lesson

Slope Review

run

riseslope

12

12

xx

yyslope

Slope Review

(7,9)

(1,1)

Find the slope of the line that passes through the points (7,9) and (1,1).

12

12

xx

yyslope

17

19

slope

6

8slope

3

4slope

The slope is positive. Note that when a line has a positive slope it rises up left to right.

Slope Review

(7,2)

(1,5)

Find the slope of the line that passes through the points (1,5) and (7,2).

12

12

xx

yyslope

71

25

slope

6

3

slope

2

1slope

The slope is negative. Note that when a line has a negative slope it falls left to right.

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

C

D

A and C are parallel; they have the same slope.

If B and D have the same length, then the trapezoid is isosceles.

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