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C HAPTER

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Midpoint Formula

Here you’ll learn how to find the halfway point between two coordinate pairs with the midpoint formula. Suppose a coordinate plane were transposed over a subway map, and the blue line went in a straight line from the point (-9, 8) to the point (1, -4). If you got on the blue line at the beginning of the line and traveled halfway to the end of the line, what would be your coordinates? How would you calculate these coordinates? In this Concept, you’ll learn how to use the midpoint formula to find the halfway point between any two coordinate pairs, such as the start point and end point of the blue line. Guidance

Consider the following situation: You live in Des Moines, Iowa and your grandparents live in Houston, Texas. You plan to visit them for the summer and your parents agree to meet your grandparents halfway to exchange you. How do you find this location? By meeting something “halfway,” you are finding the midpoint of the straight line connecting the two segments. In the above situation, the midpoint would be halfway between Des Moines and Houston. The midpoint between two coordinate pairs represents the halfway point, or the average. It is the ordered pair (xm , ym ).

(xm , ym ) = (

x1 + x2 y1 + y2 , ) 2 2

Example A

Des Moines, Iowa has the coordinates (41.59, 93.62). Houston, Texas has the coordinates (29.76, 95.36). Find the coordinates of the midpoint between these two cities. Solution: Decide which ordered pair will represent (x1 , y1 ) and which will represent (x2 , y2 ).

(x1 , y1 ) = (41.59, 93.62) (x2 , y2 ) = (29.76, 95.36) 2 y1 +y2 Compute the midpoint using the formula (xm , ym ) = ( x1 +x 2 , 2 )

41.59 + 29.76 93.62 + 95.36 , ) 2 2 (xm , ym ) = (35.675, 94.49) (xm , ym ) = (

Using Google Maps, you can meet in the Ozark National Forest, halfway between the two cities. Chapter 1. Midpoint Formula

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Example B

A segment with endpoints (9, –2) and (x1 , y1 )has a midpoint of (2, –6). Find (x1 , y1 ). Solution: Use the Midpoint Formula:

x1 +x2 2

= xm

2=

Following the same procedure:

y1 +(−2) 2

x1 + 9 → 4 = x1 + 9 2 x1 = −5

= −6 → y1 + (−2) = −12 y1 = −10

(x1 , y1 ) = (−5, −10) Example C

Find the values of x and y that make (9.5, 6) the midpoint of (3, 5) and (x, y). Solution: Start with the formula, and solve for the variables:

x1 + x2 y1 + y2 , ) 2 2 3+x 5+y (9.5, 6) = ( , ) 2 2

The midpoint formula:

(xm , ym ) = (

Substitute in the given values and variables: This can be re-written as two equations:

3+x 2 19 = 3 + x

Multiply each side by 2: Isolate the variables:

5+y 2 12 = 5 + y

9.5 =

6=

16 = x

7=y

Vocabulary

Midpoint formula: The midpoint is the ordered pair (xm , ym ) that is halfway between the points (x1 , y1 ) and (x2 , y2 ). The formula is:

(xm , ym ) =

�

x1 + x2 y1 + y2 , 2 2

�

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Guided Practice

On a hike, you and your friend decide to take different routes, but then meet up for lunch. You hike 3 miles north and 2 miles west. Starting from the same point, your friend hikes 4 miles east and 1 mile south. From these points, you each walk towards each other, meeting halfway for lunch. Where would your lunchtime meeting place be in reference to your starting point? Solution: Think of the starting point as the origin of a Cartesian coordinate system. If you walk north 3 miles, that is walking straight up the graph 3 units. Walking west 2 miles is the same as walking left 2 units on the graph. Then you have arrived at the point (-2, 3). For your friend, east is to the right (positive) and south is down (negative), so he/she arrives at (4,-1). Now you need to find the midpoint:

(xm , ym ) =

�

−2 + 4 3 + −1 , 2 2

�

=

�

2 2 , 2 2

�

= (1, 1)

Your lunch time meeting place would be 1 mile north and 1 mile east of your starting point.

Practice

Sample explanations for some of the practice exercises below are available by viewing the following videos. Note that there is not always a match between the number of the practice exercise in the videos and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. Chapter 1. Midpoint Formula

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http://www.youtube.com/watch?v=Ez_-RwV9WVo (6:41)

MEDIA Click image to the left for more content.

http://www.youtube.com/watch?v=r382kfkqAF4 (8:50)

MEDIA Click image to the left for more content.

In 1–10, find the midpoint of the line segment joining the two points. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

(x1 , y1 ) and (x2 , y2 ) (7, 7) and (–7, 7) (–3, 6) and (3, –6) (–3, –1) and (–5, –8) (3, –4) and (6, 1) (2, –3) and (2, 4) (4, –5) and (8, 2) (1.8, –3.4) and (–0.4, 1.4) (5, –1) and (–4, 0) (10, 2) and (2, –4) An endpoint of a line segment is (4, 5) and the midpoint of the line segment is (3, –2). Find the other endpoint. An endpoint of a line segment is (–10, –2) and the midpoint of the line segment is (0, 4). Find the other endpoint. 13. Shawn lives six blocks west and ten blocks north of the center of town. Kenya lives fourteen blocks east and two blocks north of the center of town. a. How far apart are these two girls “as the crow flies”? b. Where is the halfway point between their houses? Mixed Review 14. A population increases by 1.2% annually. The current population is 121,000. a. What will the population be in 13 years? b. Assuming this rate continues, when will the population reach 200,000? 15. 16. 17. 18. 19.

Write 1.29651843 · 105 in standard form. Is 4, 2, 1, 12 , 16 , 18 , . . . an example of a geometric sequence? Explain your answer. Simplify 6x3 (4xy2 + y3 z). Suppose 0√= (x − 2)(x + 1)(x − 3). What are the x−intercepts? Simplify 300.

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C HAPTER

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Distance Formula

Here you’ll learn how to find the distance between two points on a Cartesian plane. Suppose you and your friend were on a scavenger hunt. Starting out from the same place, you walked 5 blocks east and 3 blocks north. Your friend walked 7 blocks west and 2 blocks south. If each block were a tenth of a mile long, could you calculate how far apart you and your friend were? How would you do it? In this Concept, you’ll learn how to use the distance formula to determine how far two points are from each other so that you can solve this type of problem.

Guidance

To understand the distance formula, we will first look at an example:

Example A

Find the length of the segment connecting (1, 5) and (5, 2).

Solution: The question asks you to identify the length of the segment. Because the segment is not parallel to either axis, it is difficult to measure given the coordinate grid. However, it is possible to think of this segment as the hypotenuse of a right triangle. Draw a vertical line and a horizontal line. Find the point of intersection. This point represents the third vertex in the right triangle. Chapter 2. Distance Formula

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You can easily count the lengths of the legs of this triangle on the grid. The vertical leg extends from (1, 2) to (1, 5), so it is |5 − 2|= |3|= 3 units long. The horizontal leg extends from (1, 2) to (5, 2), so it is |5 − 1|= |4|= 4 units long. Use the Pythagorean Theorem with these values for the lengths of each leg to find the length of the hypotenuse.

a2 + b2 = c2 32 + 42 = c2 9 + 16 = c2 25 = c2 √ √ 25 = c2 5=c The segment connecting (1, 5) and (5, 2) is 5 units long. Mathematicians have simplified this process and created a formula that uses these steps to find the distance between any two points in the coordinate plane. If you use the distance formula, you don’t have to draw the extra lines. Distance � formula: Given points (x1 , y1 ) and (x2 , y2 ), the length of the segment connecting those two points is d = (y2 − y1 )2 + (x2 − x1 )2 . Example B

Find the distance between (–3, 5) and (4, –2). Solution: Use the distance formula. Let (x1 , y1 ) = (−3, 5) and (x2 , y2 ) = (4, −2). � � (−2 − 5)2 + (4 − (−3))2 → (−7)2 + 72 √ √ d = 98 = 7 2 units d=

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Example C

At 8 a.m. one day, Amir decides to walk in a straight line on the beach. After two hours of making no turns and traveling at a steady rate, Amir was two miles east and four miles north of his starting point. How far did Amir walk and what was his walking speed?

Solution: Plot Amir’s route on a coordinate graph. We can place his starting point at the origin, A = (0, 0). Then, his ending point will be at the point B = (2, 4). The distance can be found with the distance formula.

d=

� � √ √ (4 − 0)2 + (2 − 0)2 = (4)2 + (2)2 + 16 + 4 = 20

d = 4.47 miles.

Since Amir walked 4.47 miles in 2 hours, his speed is:

Speed =

4.47 miles = 2.24 mi/h 2 hours

Vocabulary

Distance � formula: Given points (x1 , y1 ) and (x2 , y2 ), the length of the segment connecting those two points is d = (y2 − y1 )2 + (x2 − x1 )2 . Guided Practice

Point A = (6, −4) and point B = (2, k). What is the value of k such that the distance between the two points is 5? Solution:

Use the distance formula. � � 2 2 d = (y1 − y2 ) + (x1 − x2 ) ⇒ 5 = (4 − k)2 + (6 − 2)2 Chapter 2. Distance Formula

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Square both sides of the equation. Simplify. Eliminate the parentheses.

�� �2 2 2 5 = (4 − k) + (6 − 2) 2

25 = (−4 − k)2 + 16

Simplify. Find k using the quadratic formula.

0 = k2 + 8k + 16 − 9

0 = k2 + 8k + 7 √ √ −8 ± 64 − 28 −8 ± 36 −8 ± 6 k= = = 2 2 2

k = −7 or k = −1. There are two possibilities for the value of k. Let’s graph the points to get a visual representation of our results.

Practice

Sample explanations for some of the practice exercises below are available by viewing the following videos. Note that there is not always a match between the number of the practice exercise in the videos and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. http://w ww.youtube.com/watch?v=nyZuite17Pc (9:39)

MEDIA Click image to the left for more content.

http://www.youtube.com/watch?v=T0IOrRETWhI (3:00)

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MEDIA Click image to the left for more content.

In 1–10, find the distance between the two points. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

(x1 , y1 ) and (x2 , y2 ) (7, 7) and (–7, 7) (–3, 6) and (3, –6) (–3, –1) and (–5, –8) (3, –4) and (6, 0) (–1, 0) and (4, 2) (–3, 2) and (6, 2) (0.5, –2.5) and (4, –4) (12, –10) and (0, –6) (2.3, 4.5) and (–3.4, –5.2)

11. Find all points having an x-coordinate of –4 and whose distance from point (4, 2) is 10. 12. Find all points having a y-coordinate of 3 and whose distance from point (–2, 5) is 8. 13. Michelle decides to ride her bike one day. First she rides her bike due south for 12 miles, and then the direction of the bike trail changes and she rides in the new direction for a while longer. When she stops, Michelle is 2 miles south and 10 miles west of her starting point. Find the total distance that Michelle covered from her starting point. Mixed Review 14. 15. 16. 17. 18. 19.

Solve (x − 4)2 = 121. What is the GCF of 21ab4 and 15a7 b2 ? Evaluate 10C7 and explain its meaning. Factor 6x2 + 17x + 5. Find the area of a rectangle with a length of (16 + 2m) and a width of (12 + 2m). Factor x2 − 81.

Chapter 2. Distance Formula

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C HAPTER

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Distance Formula in the Coordinate Plane

Here you’ll learn the Distance Formula and you’ll use it to find the distance between two points. What if you were given the coordinates of two points? How could you find how far apart these two points are? After completing this Concept, you’ll be able to find the distance between two points in the coordinate plane using the Distance Formula. Watch This

MEDIA Click image to the left for more content.

http://www.youtube.com/watch?v=TiAEH1qPTQU Guidance

The distance between two points (x1 , y1 ) and (x2 , y2 ) can be defined as d = the distance formula. Remember that distances are always positive! Example A

Find the distance between (4, -2) and (-10, 3). Plug in (4, -2) for (x1 , y1 ) and (-10, 3) for (x2 , y2 ) and simplify. � (−10 − 4)2 + (3 + 2)2 � = (−14)2 + (5)2 √ = 196 + 25 √ = 221 ≈ 14.87 units

d=

Example B

Find the distance between (3, 4) and (-1, 3). Plug in (3, 4) for (x1 , y1 ) and (-1, 3) for (x2 , y2 ) and simplify.

� (x2 − x1 )2 + (y2 − y1 )2 . This is called

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d= = = =

� � √ √

(−1 − 3)2 + (3 − 4)2 (−4)2 + (−1)2

16 + 1 17 ≈ 4.12 units

Example C

Find the distance between (4, 23) and (8, 14). Plug in (4, 23) for (x1 , y1 ) and (8, 14) for (x2 , y2 ) and simplify. � (8 − 4)2 + (14 − 23)2 � = (4)2 + (−9)2 √ = 16 + 81 √ = 97 ≈ 9.85 units

d=

Vocabulary

The � distance formula tells us that the distance between two points (x1 , y1 ) and (x2 , y2 ) and can be defined as d = (x2 − x1 )2 + (y2 − y1 )2 . Guided Practice

1. Find the distance between (-2, -3) and (3, 9). 2. Find the distance between (12, 26) and (8, 7) 3. Find the distance between (5, 2) and (6, 1) Answers 1. Use the distance formula, plug in the points, and simplify.

d= = = =

� (3 − (−2))2 + (9 − (−3))2 � (5)2 + (12)2 √ 25 + 144 √ 169 = 13 units

2. Use the distance formula, plug in the points, and simplify. Chapter 3. Distance Formula in the Coordinate Plane

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d= = = =

� (8 − 12)2 + (7 − 26)2 � (−4)2 + (−19)2 √ 16 + 361 √ 377 ≈ 19.42 units

3. Use the distance formula, plug in the points, and simplify. � (6 − 5)2 + (1 − 2)2 � = (1)2 + (−1)2 √ = 1+1 √ = 2 = 1.41 units

d=

Interactive Practice Practice

Find the distance between each pair of points. Round your answer to the nearest hundredth. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

(4, 15) and (-2, -1) (-6, 1) and (9, -11) (0, 12) and (-3, 8) (-8, 19) and (3, 5) (3, -25) and (-10, -7) (-1, 2) and (8, -9) (5, -2) and (1, 3) (-30, 6) and (-23, 0) (2, -2) and (2, 5) (-9, -4) and (1, -1)