Lesson 1.3 Distance and Midpoints

Geometry CP

Mrs. Mongold

Chapter 1 Foldable Flap 2

• Distance Formulas– Number Line

|| abPQ P Q

-3 4

Chapter 1 Foldable Flap 2

• Distance Formulas Cont…– Coordinate Plane

• The distance d between two points with coordinates and

),( 11 yx),( 22 yx

2122

12 yyxxd

Distance Examples

• Number Line– Use the number line to find QR

Answer

The coordinates of Q and R are –6 and –3.

QR = | –6 – (–3) | Distance Formula

= | –3 | or 3 Simplify.

Another Number Line Example

• Use the number line to find AX

Answer

• The coordinates of A and X are -5 and 3

AX = |-5 – 3| Distance Formula

AX = |-8|

AX = 8

More Distance Examples

• Coordinate Plane• Find the distance between E and F

Answer

• The coordinates are E(-4, 1) and F(3, -1)• Label them

2122

12 yyxxd

),)(,(

)1,3)(1,4(

2211 yxyx

22 11)4(3 d

22 27 d

53449 d

Another Example

• Find the distance between A and M

Answer

• The coordinates are A(-3, 4) and M(1, 2)• Label (-3, 4) and (1, 2), and

1

),( 11 yx ),( 22 yx

2122

12 yyxxd

22 42)3(1 d

22 24 d

20416 d

52d

Chapter 1 Foldable Flap 3

• Midpoint Formulas– Number Line

• The coordinate of the midpoint of a segment with endpoints that have coordinates a and b is

2

ba

Chapter 1 Foldable Flap 3

• Coordinate Plane– The coordinates of the midpoint of a segment with

endpoints that have coordinates and ),( 11 yx ),( 22 yx

2

,2

2121 yyy

xxx mm

Midpoint Examples

• The coordinates on a number line Y and O are 7 and -15 respectively. Find the coordinate of the midpoint of segment YO.

Answer

2

)15(7

2

8

4Coordinate of the midpoint of YO is -4

More Examples

• Find the coordinates of M, the midpoint of segment GH for G(8, -6) and H (-14, 12)

Answers

More Examples

• Find the coordinates of the midpoint of segment XY if X(-2, 3) and Y(-8, -9)

Answers

• (-5, -3)

What about working backwards?

• Find the coordinates of D if E(-6, 4) is the midpoint of segment DF and F(-5, -3)

Answer

• Let D be• Let E be for the midpoint• Let F be • So…..

),( 11 yx),( mm yx

),( 22 yx

221 xx

xm

2

21 yyym

)5(12 1 x )3(8 1 y

17 x 111 y

The coordinates for point D are (-7, 11)

More Examples

• What is the measure of segment PR if Q is the midpoint of segment PR?

Answer

• If Q is the midpoint then segments PQ and QR are both equal to 6 – 3x

• So, 2(6 – 3x) = 14x + 2

12 – 6x = 14x + 2

10 = 20x

x = ½

PR = 14(1/2) + 2 = 7 + 2 = 9

Last Example

• What is the measure of segment AC if B is the midpoint of segment AC?

Answer

• 2(2a – 1) = 3a + 1

4a – 2 = 3a + 1

a = 3

AC = 3(3) + 1

AC = 10

Ticket Out The Door

• Find the distance and midpoint of segment AB for A(5, 12) and B(-4, 8)