Download 1 Chapter Review

1
Chapter Review
1.1
Dynamic Solutions available at BigIdeasMath.com
Points, Lines, and Planes (pp. 3–10)
Use the diagram at the right. Give another name for plane M.
Then name a line in the plane.
h
P
You can find another name for plane M by using any three points
in the plane that are not on the same line. So, another name for
plane M is plane XYN. A line in the plane is ⃖⃗
XZ .
Z
X
Y
N
M
Use the diagram above.
1. Name a line intersecting the plane.
g
2. Name two rays.
3. Name a pair of opposite rays.
4. Name a point not in plane M.
1.2
Measuring and Constructing Segments (pp. 11–18)
a. Find EF.
57
DF = DE + EF
D
Segment Addition Postulate (Postulate 1.2)
57 = 39 + EF
Substitute 57 for DF and 39 for DE.
18 = EF
Subtract 39 from each side.
E
39
F
So, EF = 18.
— are A(6, −1) and B(3, 5). Find the distance between points A and B.
b. The endpoints of AB
Use the Distance Formula.
6
——
AB = √ (x2 − x1)2 + (y2 − y1)2
Distance Formula
——
= √ (3 −
6)2
+ [5 −
(−1)]2
4
Substitute.
—
2
= √ (−3)2 + 62
Subtract.
= √ 9 + 36
Evaluate powers.
—
—
y
B(3, 5)
= √ 45
Add.
≈ 6.7 units
Use a calculator.
2
−2
4
A(6, −1)
x
So, the distance is about 6.7 units.
Find XZ.
5.
X
17
Y
24
Z
6.
38
A
27
X
Z
Find the distance between points S and T.
7. S(−2, 4) and T(3, 9)
8. S(6, −3) and T(7, −2)
9. Plot A(8, −4), B(2, 4), C(7, 1), and D(1, −8) in a coordinate plane.
— and CD
— are congruent.
Then determine whether AB
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1.3
Using Midpoint Formulas (pp. 19–26)
— are −1 and 5. Find the coordinate of the point P that partitions
a. The endpoints of AB
—.
the segment in the ratio 2 : 1. Then find the coordinate of the midpoint M of AB
Find P. Let x1 = −1, x2 = 5, a = 1, and b = 2.
ax1 + bx2
1(−1) + 2(5)
1+2
Find M. Let x1 = −1 and x2 = 5.
x1 + x2
9
3
a+b
−1 + 5
2
4
2
—=—=—=2
— = —— = — = 3
2
—
The coordinate of the point P that partitions AB
in the ratio 2 : 1 is 3, and the coordinate of the
— is 2.
midpoint M of AB
A
−4
−2
MP
0
B
2
4
— are (2, 3) and (4, −3).
b. The endpoints of CD
Find the coordinates of the midpoint M.
C(2, 3)
2
M(?, ?)
2 + 4 3 + (−3)
M —, — = M(3, 0)
2
2
)
(
y
6
x2
x1
2
6x
4
−2
The coordinates of the midpoint M are (3, 0).
D(4, −3)
— are given. Find the coordinate of the point P that partitions the
The endpoints of AB
—.
segment in the given ratio. Then find the coordinate of the midpoint M of AB
11. −7 and 17; 2 : 1
10. 2 and 22; 1 : 3
The endpoints of a segment are given. Find the coordinates of the midpoint M.
13. G(9, −6) and H(−1, 8)
12. E(−4, 12) and F(3, 15)
—
14. The midpoint of JK is M(6, 3). One endpoint is J(14, 9). Find the coordinates of endpoint K.
—
15. Point M is the midpoint of AB where AM = 3x + 8 and MB = 6x − 4. Find AB.
1.4
Perimeter and Area in the Coordinate Plane (pp. 29–36)
Find the perimeter and area of rectangle ABCD with vertices A(−3, 4), B(6, 4), C(6, −1),
and D(−3, −1).
Draw the rectangle in a coordinate plane. Then find the length and
width using the Ruler Postulate (Postulate 1.1).
Length AB = ∣ −3 − 6 ∣ = 9
y
A
B
Width BC = ∣ 4 − (−1) ∣ = 5
2
Substitute the values for the length and width into the formulas for the
perimeter and area of a rectangle.
D
P = 2ℓ + 2w = 2(9) + 2(5) = 18 + 10 = 28
A = ℓw = (9)(5) = 45
−2
2
4
−2
x
C
So, the perimeter is 28 units, and the area is 45 square units.
Find the perimeter and area of the polygon with the given vertices.
16. W(5, −1), X(5, 6), Y(2, −1), Z(2, 6)
17. E(6, −2), F(6, 5), G(−1, 5)
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1.5
Measuring and Constructing Angles (pp. 37–46)
Given that m∠DEF = 87°, find m∠DEG and m∠GEF.
Step 1
m∠DEF = m∠DEG + m∠GEF
87° = (6x + 13)° + (2x + 10)°
87 = 8x + 23
64 = 8x
8=x
Step 2
D
Write and solve an equation to find the value of x.
(6x + 13)°
G
Angle Addition Postulate (Post. 1.4)
(2x + 10)°
Substitute angle measures.
E
Combine like terms.
F
Subtract 23 from each side.
Divide each side by 8.
Evaluate the given expressions when x = 8.
m∠DEG = (6x + 13)° = (6 ∙ 8 + 13)° = 61°
m∠GEF = (2x + 10)° = (2 ∙ 8 + 10)° = 26°
So, m∠DEG = 61°, and m∠GEF = 26°.
Find m∠ABD and m∠CBD.
18. m∠ABC = 77°
19. m∠ABC = 111°
A
A
(3x + 22)°
D
B
(−10x + 58)°
D
(5x − 17)°
B
C
(6x + 41)°
C
20. Find the measure of the angle using a protractor.
1.6
Describing Pairs of Angles (pp. 47–54)
a. ∠1 is a complement of ∠2, and m∠1 = 54°. Find m∠2.
Draw a diagram with complementary adjacent angles to
illustrate the relationship.
m∠2 = 90° − m∠1 = 90° − 54° = 36°
54°
1
2
b. ∠3 is a supplement of ∠4, and m∠4 = 68°. Find m∠3.
Draw a diagram with supplementary adjacent angles to illustrate
the relationship.
m∠3 = 180° − m∠4 = 180° − 68° = 112°
68°
4 3
∠1 and ∠2 are complementary angles. Given m∠1, find m∠2.
21. m∠1 = 12°
22. m∠1 = 83°
∠3 and ∠4 are supplementary angles. Given m∠3, find m∠4.
23. m∠3 = 116°
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24. m∠3 = 56°
Basics of Geometry
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