Midterm Exam Review
Midterm Exam Review
Midterm Exam Review
Slides 2 - 14
Warm-up (12/7 – 12/11)
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.
<5 and <16; <6 and <13; <9 and <13; <4 and <5
<7 and <2; <11 and <15; <3 and <6; <4 and 11
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
7) Given: AB = DC;
AD = BC
D Prove: ∆ABD ˜ ∆CDB
∆ZBX ˜= ∆WBY
<ZXY = <WYX
Prove: ∆ZXY ˜= ∆WYX
Points, lines and planes
9) Name a point on line m.
10) Using AI and FE,
Name a pair of acute vertical
Name a pair of obtuse vertical
11) Identify a parallel plane to
Identify all intersecting planes
to plane AVS.
12) Identify all skew lines to
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)
Distance, Midpoint and Slope Formulas
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
c) Find the slope between 2 points: (3, 9) and (14,
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.
22) Identify the polygon names,
Exterior Angle Theorem
23) Use Exterior Angle
If <3 = 80º, <6 = 108º, find <2.
If <7 = 30º, <5 = 70º, find <4.
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 number is equal to itself.
If 6 = x, then x = 6.
If x = 3 + 5 and 3 + 5 = 8, then x = 8.
If x = 3 + 5 and 8 = 3 + 5, then x = 8.
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
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.
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.
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.
Ex: Given: AB = BD
EB = BC
Prove: ∆ABE =
D S ides
C Given: CX bisects ACB
Prove: ∆ACX ˜=∆BCX
P CX bisects ACB
CX = CX
∆’s ∆ACX =˜ ∆BCX
Def of angle bisc
N – GON
SEE PAGE 57 IN
Regular or Irregular polygons
• An Equilateral polygon: all sides are
• An Equiangular polygon: all angles are
• A regular polygon: a convex polygon with
all congruent sides and angles.
• An irregular polygon: is a polygon that is
Isosceles, Equilateral Triangles
Area of a Triangle
A = ½ bh.
A = ½ (23 ft)(6 ft)
A = 69 ft 2
6 ft. Height
A = ½ bh.
Area of a Square and a Rectangle
width = 10cm
side = 8cm
length = 18cm
A = s2
A = (8cm)(8cm)
A = (18cm)(10cm)
A = 180cm2
Find L, B, S and V
L = 2rh =
B = r2
S = L + 2B
= 339.12ft2 + 2 ●113.04ft2
V = Bh
= 113.04 ● 9
Find P, B, L, S and V of a prism
Base shape is a rectangle
P = 2l + 2w
L = P●h
B = l●w
S = L + 2B
V = B●h
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