1-3 Locating Points and Midpoints Worked Out Solutions PDF

Preview

Summary

math work that is extremely helpful to all students. this is especially useful for students who are studying math....

Description

1-3 Locating Points and Midpoints Use the number line to find the coordinate of the midpoint of each segment.

18. SOLUTION: E is at –6 and L is at 11.

Find the coordinates of the midpoint of a segment with the given endpoints. 24.J(– 11.2, – 3.4), K(– 5.6, – 7.8) SOLUTION: Use the Midpoint Formula.





 The midpoint of

 The midpoint is 2.5.

is (–8.4, –5.6).

ANSWER: (–8.4, –5.6)

ANSWER: 2.5

eSolutions Manual - Powered by Cognero

Page 1

1-3 Locating Points and Midpoints Find the coordinates of the missing endpoint if B is the midpoint of 28.A(1, 7), B(–3, 1) SOLUTION: Let the coordinates of C be (x, y ). Then by the Midpoint Formula, .

 Write two equations to find the coordinates of C.

26. SOLUTION: Use the Midpoint Formula.





  The midpoint of

is

ANSWER:

 The coordinates of C are (–7, –5). ANSWER: C(–7, –5)

32. eSolutions Manual - Powered by Cognero

Page 2

1-3 Locating Points and Midpoints SOLUTION: Let the coordinates of A be (x, y ).

 Then by the Midpoint Formula, .

 Write two equations to find the coordinates of A.





 The coordinates of A are

.

ANSWER: eSolutions Manual - Powered by Cognero

Page 3

1-3 Locating Points and Midpoints

Suppose M is the midpoint of Findthemissingmeasure. 34.FM = 5y + 13, MG = 5 – 3y , FG = ?

36.FM = 8a + 1, FG = 42, a = ? SOLUTION:

SOLUTION: If M is the midpoint, then FM = MG.

  If M is the midpoint, then

 

Substitute.

So, FM = 21.



 Then y = –1.

 FM= 5y + 13 =5(–1) + 13 =8 MG = 5 – 3y = 5 – 3(– 1) =8 FG = FM + MG =8+8 =16

 ANSWER: 2.5

ANSWER: 16 eSolutions Manual - Powered by Cognero

Page 4

1-3 Locating Points and Midpoints

ANALYZE RELATIONSHIPS Refer to the number line.

41.Find the point X on

that is

of the distance from A to F.

SOLUTION:

38.Find the point X on

that is

of the distance from C to F.

SOLUTION:

The distance from A to F is 12 units. To find the point

The distance from C to F is 9 unit. To find the point of the distance from C to F, first find

of the distance from A to F, find

.

Add this distance to the coordinate of A to find that point X is at 2.6 on thenumberline.

.

or 1.8

ANSWER: 2.6

Then add this to the coordinate of C so point X is at -2.2 on the number line.

 ANSWER:

-2.2

eSolutions Manual - Powered by Cognero

Page 5

1-3 Locating Points and Midpoints

42.Find X on

that is

the distance from A to B.

SOLUTION: SOLUTION:

Since the ratio of the measure is 1:2, 2JX = XK.

Find the distance between the x-coordinates of A and B.

So, JK = JX + XK = JX + 2JX or 3JX. Thus, JX is of JK.  Find the distance between the x-coordinates of J and K.

Multiplythedistancesbythefractionaldistance. Add this to the x-coordinate of A to determine the x-coordinate of X. . The x-coordinate of X is –3.6.  Then, find the distance between the y -coordinates of A and B.

Multiplythedistancesbythefractionaldistance. Add this to the x- coordinate of J to determine the x - coordinate of X. . The x-coordinate of X is 1.  Then, find the distance between the y -coordinates of J and K.

Multiplythedistancesbythefractionaldistance. Add this to the y -coordinate of A to determine the y -coordinate of X. . The y -coordinate of X is –2.2. Thus, point X is located at (–3.6, –2.2). ANSWER: (–3.6, –2.2)

Multiplythedistancesbythefractionaldistance. Add this to the y -coordinate of J to determine the y -coordinate of X. . The y -coordinate of X is

.

Thus, point X is located at ANSWER:

44.Find X on

such that the ratio of JX to XK is 1:2.

eSolutions Manual - Powered by Cognero

Page 6

1-3 Locating Points and Midpoints 48.APPLY MATH Points A and B represent two cities. Where should the state place a rest area so it is halfway between cities A and B?

50.GEOMETRY One endpoint of has coordinates (–3, 5). If the coordinates of the midpoint of are (2, –6), what is the length of SOLUTION: First find the length of

?

.

SOLUTION: Substitute the coordinates for A and B into the midpoint formula . The distance from A to the midpoint is 12.1, thus the distance of the whole segment is . ANSWER: (1, 3)

eSolutions Manual - Powered by Cognero

ANSWER: 24.2

Page 7

1-3 Locating Points and Midpoints 54.JUSTIFY ARGUMENTS Is the point one third of the way from ( x 1 , y the point

? Explain.

SOLUTION: Sample answer: Choose some points that lie on horizontal, vertical, and di distance between the first pair of points and the first point and the new po

 (x 1, y 1) (x 2, y 2) (–3, 0)

(6, 0)

(0, 1)

(0, 13)

(0, 0)

(6, 0)

(9,12)

(0, 0)

(0, 0)

(12, 9)

(–4, –5)

(5, 7)

(3, –2)

(3, 4)

Distance between f

 Test each pair of distances. Only 2 =

and 5 =

. So when (x

way from (x 1, y 1) to (x 2, y 2). Therefore, the correct answer is sometim ANSWER:

Sample answer: sometimes; when the point (x 1, y 1) has coordinates (0,

56.Jamar plots two points, P and Q, on a coordinate plane. The midpoint of the points is M(–3, 4). Which of the following could be the points that Jamar plots? A P(–5, 10) and Q(1, 2) B P(–2, 6) and Q(–4, 2) C P(–7, 1) and Q(4, 3) D P(–1, 7) and Q(2, 3) SOLUTION: To find the endpoints of Jamar's plots, find the midpoint for each answer choice. A. This does not match the midpoint (–3, 4).  B.

This

matchesthemidpoint.So, P(–2, 6) and Q(–4,2)aretheendpoints.   C.

This

does not match the midpoint (–3, 4).  D

This

does not match the midpoint (–3, 4). ANSWER: B 60.Points A, B, C, and D are located on a number line, as shown.

Which of the following is the distance from the midpoint of midpoint of ? F  eSolutions Manual - Powered by Cognero

to the

Page 8

1-3 Locating Points and Midpoints G H J SOLUTION: Find the midpoint of

.

The midpoint of is –1.5.  Find the midpoint of .

The midpoint of is 4. The distance between the midpoints is correctchoiceisJ. 

. Thus the

ANSWER: J

eSolutions Manual - Powered by Cognero

Page 9...