Definitions and Postulates
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Transcript of Definitions and Postulates
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.