Distance Formula

Goal: to find the length of a line segment using the distance formula.

Method 1 Whenever the segments

are horizontal or vertical, the length can be obtained by counting.

When we need to find the length (distance) of a segment such as we simply COUNT the distance from point A to point B.(AB = 7)

We can use this same counting approach for .(CD = 3)

Unfortunately, this counting approach does NOT work for which is a diagonal segment.

Method 2 In the last slide, we

found the distance between two numbers on a number line. But what happens if the points are not on a straight line like points A and B above?

Method 2

• When working with diagonal segments, the Pythagorean theorem can be used to determine the length.

Method 2•Draw a right triangle using points E and F as corners.•When you have a right triangle you can use the Pythagorean theorem to find the distance.•c2=a2+b2 (Solve for c)

c a b 2 2

•Find the distance between points E and F using the Pythagorean theorem.

Work: 2 2 2a b c 2 2 23 4 x

29 16 x 225 x

225 x

25 x5 x

•The distance between E and F is 5 units.

•Use this method of working with the Pythagorean Theorem whenever you forget the Distance Formula!

Method 3

When working with diagonal segments, use the Distance Formula to determine the length. 

Now let’s use the Pythagorean theorem to develop the distance formula.

The distance between the points E = (x1, y1) and F = (x2, y2) is given by the formula: 

Length of Hypotenuse

21 2 12

2( ) ( )xd y yx

Length of Side 1 (a2)

Length of Side 2(b2)

Distance Formula:

a: Length of Side 1

b: Length of Side 2

hypotenuse/distance

E

F

Length of Hypotenuse

Length of Side 1 (a2)

Length of Side 2(b2)

21 2 12

2( ) ( )xd y yx

IMPORTANT!

Note: x2 is not the same as x2!!!!!! X2 means x - point number 2 and x2

means the square of x.

Now use the distance formula to find the

distance for the first example: Work: Coordinates of E: (1, -1) Coordinates of F: (4, -5)

2 2( ) (4 )1 1 ( 5)d

2 2( (3) )4d 169d

25d 5d

21 2 12

2( ) ( )xd y yx

Distance Formula:

It doesn't matter which point you start with.  Just start with the same point for reading both the x and y coordinates

The Distance Formula can be used to find the lengths of all forms of line segments:  horizontal, vertical and diagonal.  

The advantage of the Distance Formula is that you do not need to draw a picture to find the answer.  All you need to know are the coordinates of the endpoints of the segment.

Use the distance formula to find the distance between the following points.

1. (1,6) and (5,1)

2 2(1 5) (6 1)d 2 2( 4) (5)d

16 25d 41d

6.4d

Use the distance formula to find the distance between the following points.

2. (-5,2) and (3, -1)

2 2( 5 3) (2 ( 1))d 2 2( 8) (3)d

64 9d 73d

8.5d

The expression represents the distance formula. What are the original points?

22 )23()110( d

y2x1x2 y1

Coordinates (x2, y2) and (x1, y1)

(0, -3) and (11, 2)

Midpoint Formula

The point halfway between the endpoints of a line segment is called the midpoint.

A midpoint divides a line segment into two equal parts. In Coordinate Geometry, there are several ways to determine the midpoint of a line segment.  

Method 1:If the line segments are vertical or horizontal, you may find the midpoint by simply dividing the length of the segment by 2 and counting that value from either of the endpoints.  

Example: Find the midpoints

of line segments AB and CD.

The length of line segment CD is 3 (by counting).  The midpoint is 1.5 units from either endpoint.  On the graph, this point is (2,1.5)

•The length of line segment AB is 8 (by counting).  The midpoint is 4 units from either endpoint.  On the graph, this point is (1,4).

You are asking yourself “What point is halfway between the two?”

However, … If the line segments are diagonal, more

thought must be paid to the solution.  When you are finding the coordinates of the midpoint of a segment, you are actually finding the average (mean) of the x-coordinates and the average (mean) of the y-coordinates.

This concept of finding the average of the coordinates can be written as a formula:

The Midpoint Formula:

The midpoint of a segment with endpoints (x1 , y1) and (x2 , y2) has coordinates:

2,

22121 yyxx

"The Midpoint Formula" sung to the tune of "The Itsy Bitsy

Spider"by Halyna Reynolds

Surf City, NJ

When finding the midpoint of two points on a graph,Add the two x's and cut their sum in half.Add up the y's and divide 'em by a two,Now write 'em as an ordered pairYou've got the middle of the two.

Example:

Find the midpoint of line segment AB. A(-3,4) B(2,1)

Find the midpoint of the two points using the Midpoint formula.

1. (-2, 4) and (3, 4) 2. (8, -3) and (5, -4)

2

8,

2

1

4,5.0

2

43,

2

58

2

7,

2

13

5.3,5.6

2,

22121 yyxx

2,

22121 yyxx

Example 3: Southwestern Telephone Company uses GPS to map the

locations of its telephone poles. IT is determined that an additional pole is needed exactly halfway between 2 poles located at coordinates (20, 35) and (40, 15). What are the coordinates of the location of the new pole?

2

1535,

2

4020

2

50,

2

60 25,30

2,

22121 yyxx

4. The expression represents the midpoint formula. What are the original points?

y2x1x2 y1

Coordinates (x2, y2) and (x1, y1)

(-1, 2) and (5, 0)

sinopposit

hypote eA

e

nus

2

20,

2

15

Trigonometry: For Right Triangles Only!

leg

leg

hypotenuse - always oppositethe right angle

Basic Trigonometry Rules:

With right triangles you can use three special ratios to solve problems.

These ratios ONLY work in a right triangle.

The hypotenuse is across from the right angle.

Questions usually ask for an answer to the nearest units.

You need a scientific or graphing calculator.

Definitions:

BaC

A

bcHypotenuse

Leg a is opposite <A and adjacent to <B.

Leg b is opposite <B and adjacent to <A.

Sine

The sine (sin) of an acute angle of a right triangle is the ratio that compares the length of the leg opposite the acute angle to the length of the hypotenuse.

BaC

A

bcHypotenuse

sina

Ac

sinb

Bc

Cosine

The cosine (cos) of an acute angle of a right triangle is the ratio that compares the length of the leg adjacent the acute angle to the length of the hypotenuse.

BaC

A

bcHypotenuse

cosa

Bc

Tangent

The tangent (tan) of an acute angle of a right triangle is the ratio that compares the length of the leg opposite the acute angle to the length of the leg adjacent the acute angle .

BaC

A

bcHypotenuse

tana

Ab

tanb

Ba

Formulas: Soh Cah Toa A represents the

angle of referenceA

LegOpposite A

Hypotenuse

LegAdjacent to A

Remember:

The formulas can be remembered by:soh cah toa

The formulas can be remembered by:oscar had a heap of apples

The formulas can be remembered by:oh heck, another hour of algebra!

Examples:Find the sine, cosine and tangent ratios for both acute angles of the following right triangle.

CA

B

6

10

8

sino

Ah

8

6

cosa

Ah

tano

Aa

6

10

8

10

4

5

3

5

4

3

Examples:Find the sine, cosine and tangent ratios

for both acute angles of the following right triangle.

CA

B

6

10

8sin

oB

h

cosa

Bh

tano

Ba

6

8

6

10

8

10

4

5

3

5

3

4

Using Trig Ratio Tables/Calculator

tan 81°

Find the following values to the nearest ten-thousandth:

0.6691 0.8660cos 30°

•sin 42°

•tan 27°•sin 73° cos 36°

0.50950.9563

6.3138

0.8090

Applications: In order to determine how an object is

grown, you will need to determine the height of the object. You can use trigonometry, or the study of triangles, to find the height of an object.

The tangent function can help find the height of objects.

To determine the height of the flagpole, set up a triangle with one side being the height of the flagpole (a), another side being a distance from the flagpole to a point on the ground (b), and the third side being the distance from that point to the top of the flagpole (c). Assume the flagpole meets the ground at a right angle.

Solution: You have two options to solve this

problem: 1. Use inverse operations:

tan_

_A

measure opposite

measure adjacent

tan 458

a

8 458

8 tana

8 1 a

The height of the flagpole is 8 meters high.

8 a

Method 2: Use a proportion.

8 45 tan a

a 8 45tan

tan45

1 8

a

a 8 1

tan_

_A

measure opposite

measure adjacent

The height of the flagpole is 8 meters high.

8a

In right triangle ABC, hypotenuse AB=15 and angle A=35º.Find leg BC to the nearest tenth.

15 sin 35x

15 0.5736x

sin 35

1 15

x

8.6x

_

_

measure oppositesinA

measure hypotenuse

Example 2:

Example 3:

26.50x

In right triangle ABC, leg BC=20 and angle B = 41º.Find the hypotenuse BA to the nearest hundredth

cos 41 20x

20

cos 41x

cos 41 20

1 x

20

0.7547x

_cos

_

measure adjacentA

measure hypotenuse

cos 41 20

cos 41 cos 41

x

Example 4:

A ladder 6 feet long leans against a wall and makes an angle of 71º with the ground.  Find to the nearest tenth of a foot how high up the wall the ladder will reach.

6 sin 71x

sin 71

1 6

x

_

_

measure oppositesinA

measure hypotenuse

CA

B

6x

71

15 0.5736x 8.6x

The person in the drawing is using a hypsometer.

A hypsometer is an instrument you can use to find the height of very tall objects.

How is this example different from the one in our example 1? How will that affect the answer? What will you have to do to compensate for this

difference? Will your answer be the same?

Using a Hypsometer:

Hold the hypsometer up and look through the straw at the top of a tall object. The imaginary line that goes from your eye to the top of the object is called the line of sight. Once you have found your line of sight, look at the point at which the string crosses the semicircle. That is the measure of the angle from your line of site and an imaginary horizontal line.

Work

8 tan 45x

tan 45

1 8

x

_tan

_

measure oppositeA

measure adjacent

8 1x

8x

x

Now a = x + 2

A = 10 mHeight of person from toes to eye level.

Another Example

10 tan 60h

tan 60

1 10

h

_tan

_

measure oppositeA

measure adjacent

10 1.732h

17.32h

Now h = x + 2

A = 19.32 m

Height of person from toes to eye level.

h

Activity: You are now going to use your hypsometer to measure tall objects. Complete the data table on the back

To make hypsometer, you will need a straw, some string, a paper clip, and index cared and a protractor (paper).

Glue the semi-circle paper protractor) to the index card and cut it out.

Attach the straw to the semicircle, with the middle of the straw at the dot labeled A.

Attach a piece of string 6 inches long to the center of the straw.

Attach a paper clip to the other end of the string. Make sure that the paper clip hangs freely.

Date Table:Length of your foot (in feet):Your height from foot to eye (in feet):

Object Angle Ground Distance from Object

Tree

Building

Light Pole

Your Choice

Now use your data to calculate the height of each object. Show ALL work!