Geometry1.3 Using Midpoint and Distance Formulas

August 20, 20151.3 Using Midpoint and Distance Formulas

1.3 Essential Question:

• How is the segment length found using the midpoint

and segment bisector and how are the midpoint and

distance formulas used?

August 20, 20151.3 Using Midpoint and Distance Formulas

Midpoint

August 20, 20151.3 Using Midpoint and Distance Formulas

A M C

•M is the midpoint of segment AC.

•M bisects segment AC.

•Bisect: divide into two congruent parts.

•Congruence marks indicate congruent segments.

MCAM

August 20, 20151.3 Using Midpoint and Distance Formulas

A Segment Bisector can be a…

Segment Ray

Line Plane

Example 1

•

August 20, 20151.3 Using Midpoint and Distance Formulas

Example 2

•

August 20, 20151.3 Using Midpoint and Distance Formulas

Your Turn

Find EF.

August 20, 20151.3 Using Midpoint and Distance Formulas

August 20, 20151.3 Using Midpoint and Distance Formulas

The Coordinate Plane

x-axis

y-axis

Points are located

with a coordinate pair,

(x, y). 3

4

(3, 4)

-6

2

(-6, 2)

August 20, 20151.3 Using Midpoint and Distance Formulas

Midpoint Formula

(x1, y1)

(x2, y2)

M

2,

2

2121 yyxx

The midpoint is

the average point.

August 20, 20151.3 Using Midpoint and Distance Formulas

Example 3

Find the midpoint of segment AB if A is point (10, -6) and B is

point (2, 8).

Solution:

2

2,

2

12

2

86,

2

210(6, 1)

August 20, 20151.3 Using Midpoint and Distance Formulas

Your Turn

Find the midpoint of the segment between (20, -14) and

(-16, 4).

Solution:

2

10,

2

4

2

414,

2

)16(20(2, -5)

August 20, 20151.3 Using Midpoint and Distance Formulas

Example 4

The midpoint of segment DH is O(3, 4). One endpoint is D(5, 7).

Find the coordinates of H.

D(5,7)

O(3,4)

H(x, y)

Write out the formula:

2

7,

2

5 yx

continues…

August 20, 20151.3 Using Midpoint and Distance Formulas

Example 4 Continued

There are two equations to solve:

Solve:

1

65

32

5

x

x

x Solve:

1

87

42

7

y

y

y

2

7,

2

5 yx 4,3

continues…

August 20, 20151.3 Using Midpoint and Distance Formulas

Example 4 Continued

Since x = 1 and y = 1, the other endpoint is H(1,1).

D(5,7)

O(3,4)

H(1, 1)

August 20, 20151.3 Using Midpoint and Distance Formulas

Your Turn

The midpoint of segment AB is M(1, 5). If one endpoint is

A(-3, -4), find the coordinates of B.

Solution:

2

)4(,

2

)3( yx

Solve for x:

5

23

12

3

x

x

x Solve for y:

14

104

52

4

y

y

y

Endpoint: B(5, 14)

5,1

August 20, 20151.3 Using Midpoint and Distance Formulas

Radical Review

7 Can this be simplified? NO!

12 What about this one? YES!

3212

This is called Simplifying a Radical.4 3

2 2

August 20, 20151.3 Using Midpoint and Distance Formulas

Let’s Practice

2027 32

August 20, 20151.3 Using Midpoint and Distance Formulas

Try it. Simplify these radicals:

8200

5044

a.

d.c.

b.

August 20, 20151.3 Using Midpoint and Distance Formulas

Put down your pencils, watch, listen, and think!

August 20, 20151.3 Using Midpoint and Distance Formulas

Distances

x-axis

y-axis

(5, 3)

How far is (5, 3) from

(0, 0)?

?

5

3

This is a right

triangle.

August 20, 20151.3 Using Midpoint and Distance Formulas

The Pythagorean Theorem

a

b

c

a2 + b2 = c2

A(x1, y1) C(x2, y1)

B(x2, y2)

dy2 – y1

x2 – x1

2a2

2 1( - )x x

2 22 1 2 1( - ) + ( - ) =x x y y d

Rather than thinking through the Pythagorean Theorem to find

the distance between two points, we can use the following

reasoning to obtain the distance formula.

2+ b 2=c2

2 1+ ( - )y y 2=d

August 20, 20151.3 Using Midpoint and Distance Formulas

Distance Formula

The distance d between two points (x1, y1) and

(x2, y2) is:

2 22 1 2 1= ( - ) + ( - )d x x y y

August 20, 20151.3 Using Midpoint and Distance Formulas

August 20, 20151.3 Using Midpoint and Distance Formulas

Example 5

Find the distance between (6, 8) and (3, 4).

212

2

12 yyxxd

Does it matter which

point is chosen as

point 1 or point 2?

5

25

169

43

4836

22

22

d

x1 y1 x2 y2

August 20, 20151.3 Using Midpoint and Distance Formulas

Example 6

Find the distance between (-6, 2) and (0, 4).

102

10440

436

26

24)6(0

22

22

d

212

2

12 yyxxd

x1 y1 x2 y2

August 20, 20151.3 Using Midpoint and Distance Formulas

Your Turn

Find the distance between (9, 10) and (6, 9).

10

19

13

91069

22

22

d

x1 y1 x2 y2

212

2

12 yyxxd

August 20, 20151.3 Using Midpoint and Distance Formulas

Another Problem

172

17468

464

)2()8(

)9(735

22

22

d

Find the distance between (-5, -7) and (3, -9).x1 y1 x2 y2

212

2

12 yyxxd

Assignment

August 20, 20151.3 Using Midpoint and Distance Formulas