Definitions and Definitions and PostulatesPostulates

Definitions and Definitions and PostulatesPostulates

EXAMPLE 1 Apply the Segment Addition Postulate

Measure the length of ST to the nearest tenth of a centimeter.

SOLUTION

Align one mark of a metric ruler with S. Then estimate the coordinate of T. For example, if you align S with 2, T appears to align with 5.4.

Use Ruler Postulate.ST = 5.4 – 2 = 3.4

The length of ST is about 3.4 centimeters.ANSWER

EXAMPLE 2 Apply the the Segment Addition Postulate

SOLUTION

MapsThe cities shown on the map lie approximately in a straight line. Use the given distances to find the distance from Lubbock, Texas, to St. Louis, Missouri.

Because Tulsa, Oklahoma, lies between Lubbock and St. Louis, you can apply the Segment Addition Postulate.

LS = LT + TS = 380 + 360 = 740

The distance from Lubbock to St. Louis is about 740 miles.

ANSWER

GUIDED PRACTICE for Examples 1 and 2

Use a ruler to measure the length of the segment to the nearest inch.1

8

1.

2.

MN 1 in58

ANSWERUse ruler postulate

Use ruler postulate PQ 1 in3

8

ANSWER

GUIDED PRACTICE for Examples 1 and 2

3. Use the Segment Addition Postulate to find XZ.

In Exercises 3 and 4, use the diagram shown.

xz = xy + yz= 23 + 50= 73

Segment addition postulate

Substitute 23 for xy and 50 for yzAdd

SOLUTION

ANSWER xz = 73

GUIDED PRACTICE for Examples 1 and 2

In the diagram, WY = 30. Can you use the Segment Addition Postulate to find the distance between points W and Z?

4.

NO; Because w is not between x and z.

ANSWER

EXAMPLE 3 Find a length

Use the diagram to find GH.

Use the Segment Addition Postulate to write an equation. Then solve the equation to find GH.

SOLUTION

Segment Addition Postulate.

Substitute 36 for FH and 21 for FG.

Subtract 21 from each side.

21 + GH=36

FG + GH=FH

=15 GH

EXAMPLE 4 Compare segments for congruence

SOLUTION

Plot J(– 3, 4), K(2, 4), L(1, 3), and M(1, – 2) in a coordinate plane. Then determine whether JK and LM are congruent.

To find the length of a horizontal segment, find the absolute value of the difference of the x-coordinates of the endpoints.

Use Ruler Postulate.JK = 2 – (– 3) = 5

EXAMPLE 4 Compare segments for congruence

To find the length of a vertical segment, find the absolute value of the difference of the y-coordinates of the endpoints.

Use Ruler Postulate.LM = – 2 – 3 = 5

JK and LM have the same length. So, JK LM.=~

ANSWER

GUIDED PRACTICE for Examples 3 and 4

Use the segment addition postulate to write an equation. Then solve the equation to find WX

Use the diagram at the right to find WX.5.

vx = vw + wx144 = 37 + wx107 = wx

Segment addition postulate

Subtract 37 from each side

SOLUTION

ANSWER WX = 107

Substitute 37 for vw and 144 for vx

GUIDED PRACTICE for Examples 3 and 4

6. Plot the points A(– 2, 4), B(3, 4), C(0, 2), and D(0, – 2) in a coordinate plane. Then determine whether AB and CD are congruent.

Length of AB is not equal to the length of CD, so they are not congruent

ANSWER

EXAMPLE 3 Use the Midpoint Formula

a. FIND MIDPOINT The endpoints of RS are R(1,–3) and S(4, 2). Find the coordinates of the midpoint M.

EXAMPLE 3 Use the Midpoint Formula

252

1 + 4 2

– 3 + 2 2 =, M , – 1M

The coordinates of the midpoint M are 1,–5

2 2

ANSWER

SOLUTION

a. FIND MIDPOINT Use the Midpoint Formula.

EXAMPLE 3 Use the Midpoint Formula

FIND ENDPOINT Let (x, y) be the coordinates of endpoint K. Use the Midpoint Formula.

STEP 1 Find x.

1+ x 22

=

1 + x = 4

x = 3

STEP 2 Find y.

4+ y 12

=

4 + y = 2

y = – 2

The coordinates of endpoint K are (3, – 2).ANSWER

b. FIND ENDPOINT The midpoint of JK is M(2, 1). One endpoint is J(1, 4). Find the coordinates of endpoint K.

GUIDED PRACTICE for Example 3

3. The endpoints of AB are A(1, 2) and B(7, 8).Find the coordinates of the midpoint M.

Use the midpoint formula.

( ) 1 + 72

2 + 82

,M = M (4, 5)

ANSWER The Coordinates of the midpoint M are (4,5).

SOLUTION

GUIDED PRACTICE for Example 3

4. The midpoint of VW is M(– 1, – 2). One endpoint is W(4, 4). Find the coordinates of endpoint V.

Let (x, y) be the coordinates of endpoint V. Use the Midpoint Formula.

STEP 1 Find x.

4+ x – 12

=

4 + x = – 2

x = – 6

STEP 2 Find y.

4+ y – 22

=

4 + y = – 4

y = – 8

ANSWER The coordinates of endpoint V is (– 6, – 8)

SOLUTION

SOLUTION

EXAMPLE 4 Standardized Test Practice

Use the Distance Formula. You may find it helpful to draw a diagram.

EXAMPLE 4 Standardized Test Practice

Distance Formula

Substitute.

Subtract.

Evaluate powers.

Add.

Use a calculator to approximate the square root.

(x – x ) + (y – y )2 2 2 2 1 1 RS =

[(4 – 2)] + [(–1) –3] 2 2=

(2) + (–4 )2 2=

4+16=

20=

4.47=

The correct answer is C.ANSWER

GUIDED PRACTICE for Example 4

5. In Example 4, does it matter which ordered pair you choose to substitute for (x , y ) and which ordered pair you choose to substitute for (x , y )? Explain.

1

2

1

2

No, when squaring the difference in the coordinate you get the same answer as long as you choose the x and y value from the some period

ANSWER

GUIDED PRACTICE for Example 4

6. What is the approximate length of AB , with endpoints A(–3, 2) and B(1, –4)?

6.1 units 7.2 units 8.5 units 10.0 units

Distance Formula

Substitute.

Subtract.

(x – x ) + (y – y )2 2 2 2 1 1 AB =

[2 –(–3)] + (–4 –1) 2 2=

(5) + (5 )2 2=

Use the Distance Formula. You may find it helpful to draw a diagram.

SOLUTION

GUIDED PRACTICE for Example 4

Evaluate powers.

Add.

Use a calculator to approximate the square root.

25+25=

50=

7.2=

The correct answer is BANSWER

In the skateboard design, VW bisects XY at point T, and XT = 39.9 cm. Find XY.

Skateboard

SOLUTION

EXAMPLE 1 Find segment lengths

Point T is the midpoint of XY . So, XT = TY = 39.9 cm.

XY = XT + TY= 39.9 + 39.9= 79.8 cm

Segment Addition PostulateSubstitute.

Add.

SOLUTION

EXAMPLE 2 Use algebra with segment lengths

STEP 1 Write and solve an equation. Use the fact that that VM = MW.

VM = MW4x – 1 = 3x + 3

x – 1 = 3x = 4

Write equation.

Substitute.

Subtract 3x from each side.Add 1 to each side.

Point M is the midpoint of VW . Find the length of VM .ALGEBRA

EXAMPLE 2 Use algebra with segment lengths

STEP 2 Evaluate the expression for VM when x = 4.

VM = 4x – 1 = 4(4) – 1 = 15

So, the length of VM is 15.

Check: Because VM = MW, the length of MW should be 15. If you evaluate the expression for MW, you should find that MW = 15.

MW = 3x + 3 = 3(4) +3 = 15

GUIDED PRACTICE for Examples 1 and 2

M is midpoint and line MN bisects the line PQ at M. So MN is the segment bisector of PQ. So PM = MQ =1

78

PQ = PM + MQ

781 7

81= +

343=

Segment addition postulate.

Substitute

Add.

In Exercises 1 and 2, identify the segment bisectorof PQ . Then find PQ.

1.

SOLUTION

GUIDED PRACTICE for Examples 1 and 2

In Exercises 1 and 2, identify the segment bisector of PQ . Then find PQ.

2.

SOLUTION

M is midpoint and line l bisects the line PQ of M. So l is the segment bisector of PQ. So PM = MQ

GUIDED PRACTICE for Examples 1 and 2

STEP 2 Evaluate the expression for PQ when x = 187

PQ = 5x – 7 + 11 – 2x = 3x + 4

PQ = 3187 + 4

= 1157

Substitute for x.187

Simplify.

STEP 1 Write and solve an equation

PM = MQ

5x – 7 = 11 – 2x7x = 18

Write equation.

Substitute.

Add 2x and 7 each side.

x = 187

Divide each side by 7.