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Name

LESSON

Date

Class

Review for Mastery

Midpoint and Distance in the Coordinate Plane

1-6

The midpoint of a line segment separates the segment into two halves. You can use the Midpoint Formula to find the midpoint of the segment with endpoints G(1, 2) and H(7, 6).

y

x1 x2 y1 y2 M ______, ______ 2 2

_____ _____ M 1 7, 2 6 2 2 8 8, __ = M __ 2 2

7

H (7, 6)

M (4, 4) G (1, 2)

0 7 x

M is the midpoint _ of HG .

= M(4, 4)

Find the coordinates of the midpoint of each segment. 1.

y 6

2.

B (4, 5)

3 x

y

S ( 3, 2) 3

x 0 3 3

A ( 2, 5)

3

0

6

T (1,

4)

_

3. QR with endpoints Q(0, 5) and R(6, 7)

_

4. JK with endpoints J(1, 4) and K(9, 3)

_

Suppose M(3, 1) is the midpoint of CD and C has coordinates (1, 4). You can use the Midpoint Formula to find the coordinates of D. M (3, 1) x1 x 2 y 1 y 2 M ______, ______ 2 2 y-coordinate of D Set the coordinates equal. Replace (x1, y1) with (1, 4). Multiply both sides by 2. Subtract to solve for x2 and y2. 6).

_

x-coordinate of D 3 3 6 5 x 1 x2 ______ 2 1 x2 ______ 2 1 x2 x2

1 1 2 6

y1 y 2 ______ 2 4 y2 ______ 2 4 y2 y2

The coordinates of D are (5,

5. M( 3, 2) is the midpoint of RS , and R has coordinates (6, 0). What are the coordinates of S?

_

6. M(7, 1) is the midpoint of WX , and X has coordinates ( 1, 5). What are the coordinates of W?

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46

Holt Geometry

Name

LESSON

Date

Class

Review for Mastery

Midpoint and Distance in the Coordinate Plane

y 7

1-6

continued

B (7, 6)

The Distance Formula can be used to find the distance d between points A and B in the coordinate plane. d (x2 (7 62 36 52 7.2 x1)

2

(y2 (6

y1) 2)2

2

d

1 )2 42 16

(x1, y1) Subtract.

(1, 2); (x2, y2)

(7, 6)

0

A (1, 2)

7

x

Square 6 and 4. Add. Use a calculator.

The distance d between points A and B is the _ length of AB .

Use the Distance Formula to find the length of each segment or the distance between each pair of points. Round to the nearest tenth.

_ _

7. QR with endpoints Q(2, 4) and R( 3, 9)

8. EF with endpoints E( 8, 1) and F(1, 1)

9. T(8,

3) and U(5, 5)

10. N(4,

2) and P( 7, 1)

You can also use the Pythagorean Theorem to find distances in the coordinate plane. Find the distance between J and K. c

2

a2 5

2

b2 6

2

y

Pythagorean Theorem a 5 units and b 6 units

7

K (7, 7)

Side b is 6 units.

c a

b

x 7

25 61 c

36

Square 5 and 6. Add. Take the square root.

J (2, 1) 0

61 or about 7.8

Side a is 5 units.

Use the Pythagorean Theorem to find the distance, to the nearest tenth, between each pair of points. 11.

y 6

12.

Z (4, 5)

L ( 2, 4) y

3 x 4 0 4

Y (0, 1)

3 0 6

x 4

M (3,

4)

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47

Holt Geometry

LESSON

Practice A

Midpoint and Distance in the Coordinate Plane

LESSON

Practice B

Midpoint and Distance in the Coordinate Plane

1) and U(1, 6) and W(x 5) 2, y 3)

1-6

1-6

_

Complete the statements. 1. A coordinate plane is a plane that is divided into four regions by a horizontal number line, the 2. The location, or

Find the coordinates of the midpoint of each segment. 1. TU with endpoints T(5, .

_

(3, 2

3) 2

x-axis coordinates

, and a vertical number line, the

y-axis

2. VW with endpoints V( 2, , of a point are given by an ordered pair (x, y).

_

x y 3 __, _____ ( 4, 2)

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 47. 4. Find AB. 5. Find BC. 6. Find CA.

Use the figure for Exercises 35. The midpoint of a segment has an x-coordinate that is the average of x1 x 2 the x-coordinates of its endpoints ______ . The midpoint of a segment 2 has a y-coordinate that is the average of the y-coordinates of its y1 y2 endpoints ______ . 2 3. Q has coordinates (0, 0). R has coordinates (3, 0). _ Find the midpoint of QR .

_

26 units 26 units 4 2 units

_ _

1 1__, 0 2 (0, 1) 1 1__, 1 2

4. S has coordinates (0, 5. T has coordinates (3,

2). Find the midpoint of QS.

_

7. Name a pair of congruent segments. Find the distances.

AB and BC

2). Find the midpoint of QT .

Use the figure for Exercises 6 and 7.

_

6. I is the midpoint of HJ . H has coordinates (0, 0), and I has coordinates ( 1, 2). Sketch these points in the coordinate plane. Study the graph and guess where _ J will be. Draw HJ. 7. Find the coordinates of J by using the Midpoint Formula. Use the figure for Exercises 812. Manuel is out for a jog. The thick lines on the grid are jogging paths. He is on his way home and is at D. His home is at E. Each unit on the grid is 1 mile. 8. Name the coordinates of D. 9. Find how many miles Manuel will jog if he goes straight to the x-axis. 10. Find how many miles Manuel will jog if he stays on the jogging paths all the way home. 11. Find how many miles Manuel will jog if he goes straight to the y-axis.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

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).

6.4 units 11.4 units

( 2, 4)

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.

13.4 ft ( 1, 3)

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

3 miles 4 miles 1 mile

Holt Geometry

101.8 in.

43

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44

Holt Geometry

LESSON

Practice C

Midpoint and Distance in the Coordinate Plane

LESSON

Review for Mastery

Midpoint and Distance in the Coordinate Plane

1-6

1-6

1. When using the Distance Formula, the answer is the same regardless of which coordinates are designated (x1, y1) and (x 2, y 2). Demonstrate this fact by showing that (x1 x2)

2

2

(y1

y2)

2

(x2

x1)

2

(y2

2

y1) .

2

The midpoint of a line segment separates the segment into two halves. You can use the Midpoint Formula to find the midpoint of the segment with endpoints G(1, 2) and H(7, 6).

y

(x1

x1

2

x2)

(y1

x2

2

y2 )

y1

2

2

(x2

2y1y2 y2

x 1)

2

(y2

x2

2

y1) since

2x1x2 x1

2

2

x1 x2 y1 y2 M ______, ______ 2 2

2x1x2

y2

2

2y1y2

y1

2

_____ _____ M 1 7, 2 6 2 2 8 8 __, __ =M 2 2

7

H (7, 6)

M (4, 4) G (1, 2)

0 7 x

M is the midpoint _ of HG .

= M(4, 4) Visualize or sketch each situation. Find the answers without calculating. 2. The midpoint of a segment has coordinates (0, 0). One endpoint has coordinates (a, b). Find the coordinates of the other endpoint. 3. An endpoint of a segment has coordinates (0, 0). The midpoint has coordinates (d, e). Find the coordinates of the other endpoint. Use the figure for Exercises 47. 4. On the coordinate plane, plot points A( 3, 1), B(1, 3), C(2, _ and _ 1), _ D( 2, 1). Draw AB, AD, DC, _ and BC. Name the shape. 5. If each square on the grid represents one square meter, find the perimeter of ABCD to the nearest tenth of a meter. 6. Find the area of ABCD.

_

Find the coordinates of the midpoint of each segment. 1.

y 6

2.

B (4, 5)

3 x

y

S ( 3, 2) 3

x 0 3 3

( a,

b)

A ( 2, 5)

3

0

6

T (1,

4)

(2d, 2e) (1, 5)

_

( 1, (3, 6) (5, 0.5)

1)

3. QR with endpoints Q(0, 5) and R(6, 7)

_

4. JK with endpoints J(1, 4) and K(9, 3)

rectangle 13.4 m 10 m2 11.7 m

_

Suppose M(3, 1) is the midpoint of CD and C has coordinates (1, 4). You can use the Midpoint Formula to find the coordinates of D. M (3, 1) x1 x2 y1 y2 M ______, ______ 2 2 y-coordinate of D Set the coordinates equal. Replace (x1, y1) with (1, 4). Multiply both sides by 2. Subtract to solve for x2 and y2. 6).

_

x-coordinate of D 3 3 6 5 x 1 x2 ______ 2 1 x2 ______ 2 1 x2 x2

7. Draw BD. Find the perimeter of triangle BCD to the nearest tenth of a meter.

1 1 2 6

8. Suki found a treasure map that was laid out on a grid. T had coordinates (3, 12), U had coordinates (7, 2), and V had coordinates (13, 13). Suki read the map's instructions: First, find the midpoint of segment UV, Or the treasure forever lost will be. Label this dreadful midpoint W, Or being well-off will never trouble you. W is in line with and between T and X, X! How clichéd! Can you guess what is next? TX is exactly two times TW, Be sure at X to bring a shovel or two.

y1 y2 ______ 2 4 y2 ______ 2 4 y2 y2

The coordinates of D are (5,

Name the coordinates where Suki found the treasure.

(17, 3)

5. M( 3, 2) is the midpoint of RS , and R has coordinates (6, 0). What are the coordinates of S?

_

( 12, 4) (15, 3)

Holt Geometry

6. M(7, 1) is the midpoint of WX , and X has coordinates ( 1, 5). What are the coordinates of W? Holt Geometry

Copyright © by Holt, Rinehart and Winston. All rights reserved.

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45

46

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69

001_072_Go08an_CRF_c01.indd 46

Holt Geometry

4/11/07 3:35:26 PM

LESSON

Review for Mastery

Midpoint and Distance in the Coordinate Plane

y 7

LESSON

Challenge

Applying Midpoint and Distance

ABC for Exercises 1 4. ABC to the nearest tenth.

_ _ _

4

1-6

continued

B (7, 6)

1-6

Use

The Distance Formula can be used to find the distance d between points A and B in the coordinate plane. d (x2 (7 6

2

1. Find the perimeter of

x1) 1) 4

2

2

(y2 (6

y1) 2)

2

2

d

29.6 units

x 7

2

(x1, y1) Subtract.

(1, 2); (x2, y2)

(7, 6)

0

A (1, 2)

2. What are the midpoints of AB, BC, and CA?

( 3, 2.5), (1.5,

0.5), ( 0.5, 5)

4 0 2 4

36 52 7.2

16

Square 6 and 4. Add. Use a calculator.

The distance d between points A and B is the _ length of AB .

3. Find the perimeter of the triangle whose vertices are the midpoints of ABC. Round to the nearest tenth.

14.8 units

4. Compare the perimeter of ABC to the perimeter of the triangle whose vertices are the midpoints of ABC.

Use the Distance Formula to find the length of each segment or the distance between each pair of points. Round to the nearest tenth.

_ _

7. QR with endpoints Q(2, 4) and R( 3, 9)

8. EF with endpoints E( 8, 1) and F(1, 1)

The perimeter of ABC is twice the perimeter of the second triangle.

Use quadrilateral QRST for Exercises 5 and 6.

_ _

7.1 units

9. T(8, 3) and U(5, 5) 10. N(4,

9 units

2) and P( 7, 1)

5. Find the lengths of diagonals QS and RT. Round to the nearest tenth.

8.5 units

11.4 units

12.2; 16.1 units

_ _

3

4

You can also use the Pythagorean Theorem to find distances in the coordinate plane. Find the distance between J and K. c

2

6. Find the midpoints of QS and RT. Describe what you find.

0

4

Both midpoints are at (1, 1). This is the point where the diagonals intersect.

7. The diameter of a circle has endpoints J (12, 25) and K ( 10, 13). a. What are the coordinates of the center of the circle? b. Find the circumference and area of the circle. Use the key on your calculator and round 78.7 to the nearest tenth.

4

a2 52 25 61

b2 62 36

y

Pythagorean Theorem a 5 units and b 6 units

7

K (7, 7)

Side b is 6 units.

c a

b

x 7

Square 5 and 6. Add. Take the square root.

J (2, 1) 0

c

61 or about 7.8

Side a is 5 units.

(1, 19) units; 493.2 units2

Use the Pythagorean Theorem to find the distance, to the nearest tenth, between each pair of points. 11.

y 6

12.

Z (4, 5)

L ( 2, 4) y

3 x 4 0 4

c. Determine whether or not G(1, 1) is a point on the circle. Justify your answer.

The diameter of the circle is approximately 25.1 units, so the radius is half that distance, or about 12.55 units. The distance from the center of the circle to G is 18 units. So G is not a point on the circle.

Holt Geometry

Copyright © by Holt, Rinehart and Winston. All rights reserved.

Y (0, 1)

3 0 6

x 4

M (3,

4)

5.7 units

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9.4 units

47

48

Holt Geometry

LESSON

001_072_Go08an_CRF_c01.indd 47

Problem Solving

Midpoint and Distance in the Coordinate Plane

LESSON

4/11/07 3:54:12 PM

Reading Strategies

Compare and Contrast

Consider the following points:

1-6

1-6

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?

There are two ways to find the distance between two points in a coordinate plane. It can be found using either: 1. Distance Formula 2. Pythagorean Theorem There are two ways to find the distance between points A and B: Distance Formula d (x2 x1)2 (y2 y1)2

82.5 ft

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?

Pythagorean Theorem a2 b2 c2

d is the distance between two points. (x1, y1) and (x2, y2) are the coordinates of two points. Answer each question.

85.9 ft

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?

a and b are the legs of a right triangle. c is the hypotenuse.

1. What are the coordinates of point A on the grid above? What are the coordinates of point B? 2. Substitute the values for the coordinates of A and B into the Distance Formula. Solve for d. d (2 d (x2 x1)2 (y2 y1)2

(2, 2); ( 1, (2 ( 2))2

2)

47.4 m

( 1))2

2

18.4 m

Use the map of the Sacramento Zoo on a coordinate plane for Exercises 57. 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

Gibbons

3 0 3

(3) 9 25 5

(4) 16

2

Sacramento Zoo

Otters Hyenas

3

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

Tigers

3

Jaguars

3. Draw a right triangle in the coordinate grid, as shown. Connect points A and B. This will be the hypotenuse of your triangle. Extend a segment to the right from point B at the same time you extend a segment down from point A. These will be the other two sides of your triangle. They will continue until the segments meet. The length of the segment that you extended from point A will be a and the length of the segment that you extended from point B will be b.

4. a c2

c

4, b 3 42 3 2 16 9 25 25 5

4. Use the Pythagorean Theorem to determine the distance between points A and B.

Primates

5. Explain the difference between the Pythagorean Theorem and the Distance Formula.

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

Copyright © by Holt, Rinehart and Winston. All rights reserved.

Sample answer: The Distance Formula uses a coordinate plane. The Pythagorean Theorem uses known measures of two sides of a triangle.

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Holt Geometry

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