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Name   Class    Date    1-6

Finding Midpoints of Line Segments

Follow the steps below for each of the given line segments.

A Usearulertomeasurethelengthofthelinesegmenttothenearestmillimeter.

B Findhalfthelengthofthesegment.Measurethisdistancefromoneendpointtolocatethemidpointofthesegment.Plotapointatthemidpoint.

C Recordthecoordinatesofthesegment’sendpointsandthecoordinatesofthesegment’smidpointinthetablebelow.

D Ineachrowofthetable,comparethex-coordinatesoftheendpointstothex-coordinateofthemidpoint.Thencomparethey-coordinatesoftheendpointstothey-coordinateofthemidpoint.Lookforpatterns.

REFLECT

1a. Makeaconjecture:Ifyouknowthecoordinatesoftheendpointsofalinesegment,howcanyoufindthecoordinatesofthemidpoint?

1b. Whatarethecoordinatesofthemidpointofalinesegmentwithendpointsattheoriginandatthepoint(a,b)?

E X p l o r E1

Midpoint and Distance in the Coordinate PlaneGoing DeeperEssential question: How can you find midpoints of segments and distances in the coordinate plane?

Endpoint EndpointCoordinates

Endpoint EndpointCoordinates

Midpoint Coordinates

A B

C D

E F

G-GPE.2.6

Chapter 1 31 Lesson 6

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The patterns you observed can be generalized to give a formula for the coordinates of the midpoint of any line segment in the coordinate plane.

Using the Midpoint Formula

___

PQ has endpoints P(−4, 1) and Q(2, −3). Prove that the midpoint M of ___

PQ lies in Quadrant III.

A Use the given endpoints to identify x 1 , x 2 , y 1 , and y 2 .

x 1 = -4, x 2 = 2, y 1 = , y 2 =

B By the midpoint formula, the x-coordinate of M is x 1 + x 2

______ 2 = -4 + 2 ______ 2 = -2 ___ 2 = -1.

The y-coordinate of M is y 1 + y 2

______ 2 = +

____________ 2 = _____ 2 = .

M lies in Quadrant III because

REFLECT

2a. What must be true about PM and QM? Show that this is the case.

Finding a Distance in the Coordinate Plane

You can use the Pythagorean Theorem to help you find the distance between the points A(2, 5) and B(−4, −3).

A Plot the points A and B in the coordinate plane at right.

B Draw ___

AB.

C Draw a vertical line through point A and a horizontal line through point B to create a right triangle. Label the intersection of the vertical line and the horizontal line as point C.

D Each small grid square is 1 unit by 1 unit. Use this fact to find the lengths AC and BC.

AC = BC =

E X A M P L E2

The Midpoint Formula

The midpoint M of ___

AB with endpoints A( x 1 , y 1 ) and B( x 2 , y 2 ) is given by

M ( x 1 + x 2 ______ 2 ,

y 1 + y 2 ______ 2 ) .

G-GPE.2.4

E X P L o r E3prep for G-GPE.2.4

Chapter 1 32 Lesson 6

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E By the Pythagorean Theorem, AB2 = AC2 + BC2 .Complete the following using the lengths from Step D.

AB 2 =

2

+

2

F Simplify the right side of the equation. Then solve for AB.

AB 2 = , AB =

REFLECT

3a. Explain how you solved forAB in Step F.

3b. Can you use the above method to find the distance between any two points in the coordinate plane? Explain.

The process of using the Pythagorean Theorem can be generalized to give a formula for finding the distance between two points in the coordinate plane.

UsingtheDistanceFormula

Prove that ___

CD is longer than ___

AB.

A Write the coordinates of A,B,C, and D.

A(−2, 3), B , C , D

B Use the distance formula to find AB and CD.

AB = √ [ 4 - (-2) ] 2 + (2 - 3 ) 2 = √ 6 2 + (-1 ) 2 = √ 36 + 1 = √ 37

CD = √ ( - )

2

+ ( - ) 2

= √ ( )

2

+ ( ) 2

= √

So, ___

CD is longer than ___

AB because

REFLECT

4a. When you use the distance formula, does the order in which you subtract the x-coordinates and the y-coordinates matter? Explain.

The Distance Formula

The distance between two points ( x1 , y1 ) and ( x2 , y2 ) in the coordinate plane is

√ ( x2 - x1 ) 2 + ( y2 - y1 ) 2 .

E X A M P L E4G-GPE.2.4

Chapter 1 33 Lesson 6

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1. Findthecoordinatesofthemidpointof___

ABwithendpointsA(-10,3)andB(2,-2).

2. ___

RShasendpointsR(3,5)andS(−3,−1).ProvethatthemidpointMof___

RSlieson they-axis.

3. ___

CDhasendpointsC(1,4)andD(5,0).___

EFhasendpointsE(4,5)andF(2,-1).Provethatthesegmentshavethesamemidpoint.

Use the figure for Exercises 4−6.

4. FindthedistancebetweenJandL.

5. Findthelengthof____

LM.

6. ProvethatJK=KL.

7. ___

GHhasendpointG(2,7)andmidpointM(5,1).WritetwoequationsyoucanusetofindthecoordinatesoftheendpointH.ThensolvetheequationsandwritethecoordinatesofH.

8. Asegment____

ABhasendpointsA(x1,y1)andB(x2,y2).Thesegmentisreflectedacrossthex-axis.Findthecoordinatesofthereflectedsegment.Thenshowthelengthofthereflectedsegmentisthesameasthelengthoftheoriginalsegment.Explainyourreasoning.

p r a c t i c e

Chapter 1 34 Lesson 6

Name ________________________________________ Date __________________ Class__________________

Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.

Holt McDougal Geometry

Practice Midpoint and Distance in the Coordinate Plane

Find the coordinates of the midpoint of each segment.

1. TU with endpoints T(5, −1) and U(1, −5) _______________

2. VW with endpoints V(−2, −6) and W(x + 2, y + 3) _______________

3. Y is the midpoint of XZ . X has coordinates (2, 4), and Y has coordinates (–1, 1). Find the coordinates of Z. _______________

Use the figure for Exercises 4–7.

4. Find AB. _______________

5. Find BC. _______________

6. Find CA. _______________

7. Name a pair of congruent segments. _______________

Find the distances. 8. Use the Distance Formula to find the distance, to the

nearest tenth, between K(−7, −4) and L(−2, 0). _______________ 9. Use the Pythagorean Theorem to find the distance, to

the nearest tenth, between F(9, 5) and G(–2, 2). _______________

Use the figure for Exercises 10 and 11. Snooker is a kind of pool or billiards played on a 6-foot-by-12-foot table. The side pockets are halfway down the rails (long sides). 10. Find the distance, to the nearest tenth of a foot, diagonally

across the table from corner pocket to corner pocket.

_________________________________________

11. Find the distance, to the nearest tenth of an inch, diagonally across the table from corner pocket to side pocket.

_________________________________________

6

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1-6Name   Class    Date   

Additional Practice

Chapter 1 35 Lesson 6

Name ________________________________________ Date __________________ Class__________________

Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.

Holt McDougal Geometry

Problem Solving Midpoint and Distance in the Coordinate Plane

For Exercises 1 and 2, use the diagram of a tennis court. 1. A singles tennis court is a rectangle 27 feet wide

and 78 feet long. Suppose a player at corner A hits the ball to her opponent in the diagonally opposite corner B. Approximately how far does the ball travel, to the nearest tenth of a foot?

_________________________________________

2. A doubles tennis court is a rectangle 36 feet wide and 78 feet long. If two players are standing in diagonally opposite corners, about how far apart are they, to the nearest tenth of a foot?

_________________________________________

A map of an amusement park is shown on a coordinate plane, where each square of the grid represents 1 square meter. The water ride is at (−17, 12), the roller coaster is at (26, −8), and the Ferris wheel is at (2, 20). Find each distance to the nearest tenth of a meter. 3. What is the distance between the water

ride and the roller coaster?

_________________________________________

4. A caricature artist is at the midpoint between the roller coaster and the Ferris wheel. What is the distance from the artist to the Ferris wheel?

________________________________________

Use the map of the Sacramento Zoo on a coordinate plane for Exercises 5–7. Choose the best answer. 5. To the nearest tenth of a unit, how far is it from

the tigers to the hyenas? A 5.1 units C 9.9 units B 7.1 units D 50.0 units

6. Between which of these exhibits is the distance the least? F tigers and primates G hyenas and gibbons H otters and gibbons J tigers and otters

7. Suppose you walk straight from the jaguars to the tigers and then to the otters. What is the total distance to the nearest tenth of a unit? A 11.4 units C 13.9 units B 13.0 units D 14.2 units

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Problem Solving

Chapter 1 36 Lesson 6