Transcript Midterm Exam Review

Midterm Exam Review
34 questions
Slides 2 - 14
Geometry Honors
Mrs. Tiesi
Warm-up (12/7 – 12/11)
Chapter 3
1) Write an equation in slope-intercept form for a line perpendicular to
a) y = (3/4)x + 2 containing (6, 5).
b) y = (1/3)x + 2 containing (8, 9).
2) Determine whether lines AB and CD are parallel, perpendicular, or neither.
a) A(1, 5), B(4, 4), C(9, -10), D(-6, -5)
b) A(4, 2), B(-3, 1), C(6, 0), D(-10, 8)
3) Find the distance between each pair of parallel lines with the given equations.
a) y = 5x - 22 and y = 5x + 4
b) y = (1/3)x - 3 and y = (1/3)x + 2
Hint: Find the slope-intercept equation of line p (perpendicular to line a at the same y-intercept (0, b)). Solve a sytem of equations (to find an
intercepted point (x, y)) using the equations of lines b and p. Use the Distance Formula for 2 points (0, b) and (x, y) to find the answer.
4) Given the following information, determine which lines, if any, are parallel. State the postulate or
theorem that justifies your answer.
a)
<5 and <16; <6 and <13; <9 and <13; <4 and <5
b)
<7 and <2; <11 and <15; <3 and <6; <4 and 11
m
n
d
c
Chapters 1 & 4
2-column & Flow proof
5) Sec 2-6: Write a 2-column proof
a) If -4(x - 3) + 5x = 24, then x = 12.
b) If (8 – 3x) / 4 = 32, then x = -40.
6) (Sec 4-4 and 4-5) Write a flow proof
a) Given: BA = DC, <BAC = <DCA
Prove: BC = DA
b) Given: R is the midpoint of QS,
<PQR = <TSR
Prove: ∆PRS = ∆TRS
A
B
D
C
S
P
R
Q
T
B
C
7) Given: AB = DC;
AD = BC
=
D Prove: ∆ABD ˜ ∆CDB
A
X
Y
B
Z
8) Given:
∆ZBX ˜= ∆WBY
<ZXY = <WYX
Prove: ∆ZXY ˜= ∆WYX
W
Points, lines and planes
9) Name a point on line m.
B
10) Using AI and FE,
Name a pair of acute vertical
angles
Name a pair of obtuse vertical
angles
F
H
D
C
A
G
E
m
H
V
11) Identify a parallel plane to
plane QWM.
Identify all intersecting planes
to plane AVS.
12) Identify all skew lines to
AV.
I
S
A
W
Q
M
P
Volumes/Surface Areas of all solids
13) Baseballs and softballs come in different sizes for
different types of leagues. If the diameter of a baseball is
5 inches and a softball has a diameter of 5.4 inches, find
the difference between the volumes of the two balls.
Round to the nearest tenth (V = 4πr3/3).
14) Cakes are stacked in 2 layers as a cylinder. If the radius
of a cake is 10 inches and its height is 12 inches, find the
surface area of the frosting (T = 2πrh + 2πr2).
15) A rectangular prism pool needs to be painted. If the
bottom of the pool has dimensions 15 ft x 20 ft, and its
height is 8ft, find the paint area. (Paint Area = Ph + B ;
P = 2l + 2w ; B = l ● w)
Chapter 1
Distance, Midpoint and Slope Formulas
16)
a) A segment has a midpoint at (3, 9) and an
endpoint at (14, -8), what is the other endpoint?
b) Find the distance between 2 points: (3, 9) and
(14, -8).
c) Find the slope between 2 points: (3, 9) and (14,
-8)
Triangle Problems
17) Use Angle Sum Theorem to find <A of ∆ABC, given <B = 750 and
<C = 350.
18) Classify the triangle by its angles.
<A = 10º, <B = 160º, <C = 10º
<A = 60º, <B = 60º, <C = 60º
<A = 70º, <B = 50º, <C = 60º
<A = 60º, <B = 30º, <C = 90º
19) Classify the triangle by its sides.
AB = 1/3 cm , BC = 1/3 cm, AC = ½ cm
AB = 8 in, BC = 10 in, AC = 12 in
AB = 2.1 ft, BC = 2.1 ft, AC = 2.1ft
20) ∆ABC is an isosceles right triangle. <B is the vertex right angle. If AB = 4x –
1, BC = 2x + 5 and AC = 8x – 8.44, find x and all sides. Also, what is the
measurement of <A or <C?
21) ∆ABC is an equilateral triangle. If <A = 4x, solve for x. If AB = 3y – 10, BC
= 2y + 5, and AC = y + 20, find y and all sides.
Chapter 1
Polygons
22) Identify the polygon names,
concave/convex, regular/irregular
Chapter 4
Exterior Angle Theorem
23) Use Exterior Angle
1
Theorem,
If <3 = 80º, <6 = 108º, find <2.
If <7 = 30º, <5 = 70º, find <4.
3 5
2
4 6
7
8
The rest of the problems
24) Given <1 and <2: State the theorems/postulates (Complement, Supplement, Vertical,
Corresponding, Alternate Interior, Alternate Exterior, Consecutive Interior angles) for each
of the following statement:
<1 = <2
<1 + <2 = 90º
<1 + <2 = 180º
25) Find the Volume and Surface Area of a cylinder, given r = 10cm, h = 8cm.
26) Find the Volume and Surface Area of a rectangular prism, given length = 11cm, width = 7cm
and height = 5cm.
27) Find the base of a triangle, given h = 10cm and A = 200cm2.
28) Find the width of a rectangle, given w = 11 in and A = 132 in2.
29) Find the diagonal of a square, given A = 100cm2
30) Write a converse, inverse and contrapositive of the following statement: If you are a dancer,
then you love to dance.
31) Identify the number properties:
a)
A number is equal to itself.
b)
If 6 = x, then x = 6.
c)
If x = 3 + 5 and 3 + 5 = 8, then x = 8.
d)
If x = 3 + 5 and 8 = 3 + 5, then x = 8.
Chapter 4
32)
a) Position and label an isosceles triangle
ABC with a base BC of 2a.
b) Position and label an equilateral triangle
ABC with a side of 2a.
33) Prove 3 points form a right
triangle
P(0, 0), N(3b, 0), M(0, 2a)
• Find the slopes of all 3 sides.
• Check if 2 sides form perpendicular lines.
– 1 slope is zero and the other is undefined.
– 1 slope is the reverse reciprocal of the other.
Chapter 9
34) Given an original point of (2, -4)
a) Reflection over the x-axis, y-axis, y = x, y
= 1, x = -2.
b) Translation using a component vector <3, 2>
c) Rotation 90, 180 and 270
counterclockwise about the origin.
Notes
Chapter 4
Proving triangles congruent
a) 5 methods:
Congruent Triangles: all corresponding angles
and sides are congruent.
SSS: 3 pairs of corresponding sides
SAS: 2 pairs of corresponding sides, the included
angle is between 2 sides.
ASA: 2 pairs of corresponding angles, the included
side is between the 2 angles.
AAS: 2 angles and 1 non-included side.
b) CPCTC: Corresponding parts of congruent
triangles are congruent.
A
C
B
1 2
E
SAS
Ex: Given: AB = BD
EB = BC
Prove: ∆ABE =
˜ ∆DBC
Our Outline
P rerequisites
D S ides
A ngles
S ides
Triangles =˜
C Given: CX bisects ACB
A
˜
B
=
12
Prove: ∆ACX ˜=∆BCX
AAS
A
X
B
P CX bisects ACB
A
1= 2
A
A= B
S
CX = CX
∆’s ∆ACX =˜ ∆BCX
Given
Def of angle bisc
Given
Reflexive Prop
AAS
POLYGON NAMES
NAMES
TRIANGLE
QUADRILATERAL
PENTAGON
HEXAGON
HEPTAGON
OCTAGON
NONAGON
DECAGON
HENDECAGON
DODECAGON
N – GON
#SIDES
3
4
5
6
7
8
9
10
11
12
N
SEE PAGE 57 IN
TEXTBOOK
CONCAVE
CONVEX
Regular or Irregular polygons
• An Equilateral polygon: all sides are
congruent.
• An Equiangular polygon: all angles are
congruent.
• A regular polygon: a convex polygon with
all congruent sides and angles.
• An irregular polygon: is a polygon that is
not regular.
Isosceles, Equilateral Triangles
vertex angle
A
B
leg
leg
base angles
A
base
C
B
C
Area of a Triangle
A = ½ bh.
A = ½ (23 ft)(6 ft)
A = 69 ft 2
6 ft. Height
Base
23 ft.
A = ½ bh.
Area of a Square and a Rectangle
width = 10cm
side = 8cm
length = 18cm
A = s2
A=l●w
A = (8cm)(8cm)
A=
64cm2
A = (18cm)(10cm)
A = 180cm2
Find L, B, S and V
L = 2rh =
2(3.14)(6)(9)
6ft
= 339.12ft2
B = r2
= 3.14(6)2
= 113.04ft2
9ft
S = L + 2B
= 339.12ft2 + 2 ●113.04ft2
= 565.2ft2
V = Bh
= 113.04 ● 9
= 1017.4ft3
Find P, B, L, S and V of a prism
Base shape is a rectangle
P = 2l + 2w
L = P●h
5cm
B = l●w
S = L + 2B
3cm
4cm
V = B●h
Important Properties
•
•
•
•
Reflexive : a = a
Symmetric : if a = b then b = a
Transitive : if a = b and b = c then a = c
Substitution : if a = b then a may be used in
any equation instead of b
• Distributive: a(b + c) = ab + ac
• Addition and Subtraction
if a = b then a+c = b+c and a-c = b-c
• Multiplication and Division
if a = b then ac = bc and a / c = b / c