Name
Date
Class
Reteach
LESSON
1-6
Midpoint and Distance in the Coordinate Plane
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
M 1 2 7, 2 2 6
y2 x1 x2 y______ M ______ , 1 2 2
_____ _____
2 2
(
M is the midpoint _ of HG .
-
8 8 , __ = M __
'
= M(4, 4)
X
Find the coordinates of the midpoint of each segment. 1.
2.
Y
!
Y
3 "
X
X
4
(⫺1, ⫺1)
(1, 5) _
3. QR with endpoints Q(0, 5) and R(6, 7) _
4. JK with endpoints J(1, –4) and K(9, 3)
(3, 6) (5, ⫺0.5)
_
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.
y2 x1 x2 y______ M (3, 1) M ______ , 1 2 2
x-coordinate of D x1 x2 3 ______ 2 1x 3 ______2 2 6 1 x2 5 x2
y-coordinate of D
Multiply both sides by 2.
y1 y2 1 ______ 2 4y 1 ______2 2 2 4 y2
Subtract to solve for x2 and y2.
6 y2
Set the coordinates equal. Replace (x1, y1) with (1, 4).
The coordinates of D are (5, 6).
_
5. M(3, 2) is the midpoint of RS , and R has coordinates (6, 0). What are the coordinates of S?
(⫺12, 4)
_
6. M(7, 1) is the midpoint of WX , and X has coordinates (1, 5). What are the coordinates of W? Copyright © by Holt, Rinehart and Winston. All rights reserved.
46
(15, ⫺3) Holt Geometry
Name
Date
Class
Reteach
LESSON
1-6
Midpoint and Distance in the Coordinate Plane
The Distance Formula can be used to find the distance d between points A and B in the coordinate plane.
continued
Y
" "
2 2
d
(x2 x1)
(y2 y1)
(7 1 )
(6 2)
2 2
D
(x1, y1) (1, 2); (x2, y2) (7, 6)
2 2
6 4
Subtract.
36 16
The distance d between points A and B is_ the length of AB .
Square 6 and 4.
52
Add.
7.2
Use a calculator.
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)
7.1 units
9 units
9. T(8, 3) and U(5, 5)
10. N(4, 2) and P(7, 1)
8.5 units
11.4 units
You can also use the Pythagorean Theorem to find distances in the coordinate plane. Find the distance between J and K. 2 2 2 c a b 2
5 6
Y
Pythagorean Theorem
2
+
a 5 units and b 6 units
25 36
Square 5 and 6.
61
Add.
c 61 or about 7.8
C *
Side b is 6 units.
B A
Take the square root.
Side a is 5 units.
Use the Pythagorean Theorem to find the distance, to the nearest tenth, between each pair of points. 11.
12.
Y
, Y
:
X
9
X
5.7 units Copyright © by Holt, Rinehart and Winston. All rights reserved.
-
9.4 units 47
Holt Geometry
Name
Date
Class
Reteach
LESSON
3-5
Slopes of Lines
The slope of a line describes how steep the line is. You can find the slope by writing the ratio of the rise to the run. run: go right 6 units 3 ⫽ __ rise ⫽ __ 1 Y slope ⫽ ____ run 6 2 " rise: go up 3 units You can use a formula to calculate the slope m of the line through points (x1, y1) and (x2, y2). ! Change in y-values ⫺ y y rise 2 1 ______ m = ____ run = x2 ⫺ x1 X Change in x-values
‹__›
To find the slope of AB using the formula, substitute (1, 3) for (x1, y1) and (7, 6) for (x2, y2).
y2 ⫺ y1 m = ______ x2 ⫺ x1 ⫺3 _____ =6 7⫺1 3 = __ 6 1 = __ 2
Slope formula Substitution Simplify. Simplify.
Use the slope formula to determine the slope of each line. Y
Y
(
#
$
2
X 0
2
2
2
X
2
0
*
2
‹___›
‹__›
1. HJ
2. CD
2 __ 3
0
Y 2
,
Y
2 X
0 2
2
3
X 0
2
3
2
‹__›
‹__›
3. LM
4. RS
4 __ 3
2 Copyright © by Holt, Rinehart and Winston. All rights reserved.
38
Holt Geometry
Name LESSON
3-5
Date
Class
Reteach Slopes of Lines
continued
Slopes of Parallel and Perpendicular Lines Y
Y
,
.
2
2
X 2
0
2
‹__›
slope of LM = ⫺3 ‹__›
slope of NP = ⫺3 0
4 -
. ‹__›
slope of NP ⫽ ⫺3 0 4 2 ‹___› 1 1 slope of QR ⫽ __ 2 3 0 product of slopes: 1 ⫽ ⫺1 ⫺3 __ 3 Perpendicular lines have slopes that are opposite reciprocals. The product of the slopes is ⫺1. 2
X
Parallel lines have the same slope.
Use slopes to determine whether each pair of distinct lines is parallel, perpendicular, or neither. ‹___›
‹__›
3 6. slope of EF ⫽ ⫺ __ 4 ‹__› 3 slope of CD ⫽ ⫺ __ 4
5. slope of PQ ⫽ 5 ‹__›
1 slope of JK ⫽ ⫺ __ 5
perpendicular
parallel
‹___›
‹__›
5 7. slope of BC ⫽ ⫺ __ 3 ‹__› 3 __ slope of ST ⫽ 5
1 8. slope of WX ⫽ __ 2 ‹__› 1 slope of YZ ⫽ ⫺ __ 2
perpendicular
neither
Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. Y
Y
X
X
‹__›
‹__›
‹__›
9. FG and HJ for F (–1, 2), G(3, –4), H(–2, –3), and J(4, 1)
perpendicular Copyright © by Holt, Rinehart and Winston. All rights reserved.
‹__›
10. RS and TU for R (–2, 3), S(3, 3), T (–3, 1), and U(3, –1)
neither 39
Holt Geometry
NAME
DATE
3-3
PERIOD
Skills Practice Slopes of Lines
Determine the slope of the line that contains the given points. 1. S(-1, 2), W(0, 4)
2. G(-2, 5), H(1, -7)
3. C(0, 1), D(3, 3)
4. J(-5, -2), K(5, -4)
Find the slope of each line. y
5.
y
6. 1
x
0
5 8
x
0
/
⎯⎯ and MN ⎯⎯ are parallel, perpendicular, or neither. Determine whether AB Graph each line to verify your answer.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
7. A(0, 3), B(5, -7), M(-6, 7), N(-2, -1) 9. A(-2, -7), B(4, 2), M(-2, 0), N(2, 6)
8. A(-1, 4), B(2, -5), M(-3, 2), N(3, 0) 10. A(-4, -8), B(4, -6), M(-3, 5), N(-1, -3)
Graph the line that satisfies each condition. 11. slope = 3, passes through A(0, 1)
3 , passes through R(-4, 5) 12. slope = - − 2
y
O
y
x
13. passes through Y(3, 0), parallel to DJ with D(-3, 1) and J(3, 3)
O
14. passes through T(0, -2), perpendicular with C(0, 3) and X(2, -1) to CX
y
O
Chapter 3
x
y
x
O
35
x
Glencoe Geometry
