geometry - mpsaz.org · midpoint 1.3 using midpoint and distance formulas august 19, 2016 a m c...

Geometry1.3 Using Midpoint and Distance Formulas

August 19, 20161.3 Using Midpoint and Distance Formulas

Essential Question

How can you find the midpoint and

length of a line segment in a

coordinate plane?

August 19, 20161.4 Perimeter and Area in the Coordinate Plane

What You Will Learn

• Find segments lengths using midpoints and segment

bisectors.

• Use the Midpoint Formula.

• Use the Distance Formula.


Midpoint


A M C

•M is the midpoint of segment AC.

•M bisects segment AC.

•Bisect: divide into two congruent parts.

•Congruence marks (Tic marks) indicate

congruent segments.

MCAM


A Segment Bisector can be a…

Segment Ray

Line Plane

Example 1

•


Example 2

•


Your Turn

Find EF.



The Coordinate Plane

x-axis

y-axis

Points are located

with a coordinate pair,

(x, y).3

4

(3, 4)

-6

2

(-6, 2)


Midpoint Formula

(x1, y1)

(x2, y2)

M

2,

2

2121 yyxxM

The midpoint is

the average point.


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)


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)


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:

)4,3(2

7,

2

5

yx

continues…


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…


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)


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


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


Let’s Practice

2027 32


Try it. Simplify these radicals:

8200

5044

a.

d.c.

b.


1.3 Day 2 Warm up

200 216a. b.

Simplify these radicals:


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


Distances

x-axis

y-axis

(5, 3)

How far is (5, 3) from

(0, 0)?

?

5

3

This is a right

triangle.


The Pythagorean Theorem

a

b

c

a2 + b2 = c2


Distances

x-axis

y-axis

(5, 3)

How far is (5, 3) from

(0, 0)?

?

5

3

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


Distance Formula

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

(x2, y2) is:

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



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

x1y2x2 y1


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


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


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

Essential Question

How can you find the midpoint and

length of a line segment in a

coordinate plane?

August 19, 20161.4 Perimeter and Area in the Coordinate Plane

Assignment

1.3 Day 1

#1 – 9 odd, 15 – 21 odd, 46 – 53

1.3 Day 2

1.3 ws all (#22-24 do midpt)


geometry - mpsaz.org · midpoint 1.3 using midpoint and distance formulas august 19, 2016 a m c...

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