Download Practice Quiz Key

Practice Quiz
Polygons, Area
Perimeter, Volume
1 Two angles of a hexagon measure 140° each. The other
four angles are equal in measure. What is the measure
of each of the other four equal angles, in degrees?
x
x
Step 1: Find the sum of interior
angles in a hexagon.
 Number of sides = 6
140
x
140
 Number of sides – 2
=6–2=4
x
 Multiply 4 by 180
= 4(180) = 720
1 Two angles of a hexagon measure 140° each. The other
four angles are equal in measure. What is the measure
of each of the other four equal angles, in degrees?
x
140
x
140
Step 1: Find the sum of interior
angles in a hexagon. 720
Step 2: Set up equation by
letting sum of angles equal 720.
x + x + x + x + 140 + 140 = 720
4x + 280 = 720
x
x
– 280 – 280
4x
= 440
Measure of each the four
equal angles is 110
x
= 110
2
In trapezoid ABCD, AB = CD.
What is the value of x?
Method #1
Isosceles Trapezoid
A and B
supplementary
75° + x = 180°
–75
–75
x = 105°
2
In trapezoid ABCD, AB = CD.
What is the value of x?
Method #2
Sum of all
angles = 360°
x + x + 75 + 75 = 360
2x + 150 = 360
–150 –150
2x = 210
x = 105
3
In quadrilateral DEFG, DG is parallel to
EF . What is the measure of F?
F and G
supplementary
x + 10 + x = 180
2x + 10 = 180
–10 –10
2x
= 170
x = 85
Trapezoid
E
F
(x+10)
D
x
G
F = x + 10
= 85 + 10 = 95
4
In the figure, what is the value of x ?
Step 1: Solve for c using pythagorean theorem.
a2 + b2 = c2
3 
2
 7
2
c
3
2
3
c
c2
9 + 7 =
16 = c2
16  c
4 =c
2
x
7
4
In the figure, what is the value of x ?
Step 2: Solve for x using pythagorean theorem.
a2 + b2 = c2
32 + 42 = x2
x2
9 + 16 =
25 = x2
25  x
5 =x
3
3
4
x
2
7
5
In the figure, what is the length of AB ?
Step 1: Solve for ? using pythagorean theorem.
A
2
2
2
a +b =c
4
32 + 42 = ?2 D
x
?
9 + 16 = ?2
3
25 = ?2
25  ?
5 =?
C
13
2
x is length of AB
B
5
In the figure, what is the length of AB ?
Step 2: Solve for x using pythagorean theorem.
A
2
2
2
a +b =c
4
2
2
2
x + 5 = 13
D
x
5
x2 + 25 = 169
3
–25 –25
x2
= 144
x  144
C
13
2
x = 12
x is length of AB
B
6
Find the value of each interior angle
for a regular polygon with 20 sides.
Step 1: Find sum of the interior angles in a
regular polygon with 20 sides.
 Number of sides = 20
 Number of sides – 2 = 20 – 2 = 18
 Multiply 18 by 180 = 18(180) = 3240
Step 2: Find value of each interior angle.
Divide sum by number of sides, 20.
3240  20 = 162
7
A regular octagon is shown.
What is the measure, in degrees, of X?
Step 1: Find sum of the interior angles in
the regular octagon.
 Number of sides = 8
X
 Number of sides – 2 = 8 – 2 = 6
 Multiply 6 by 180
= 6(180) = 1080
Step 2: Find value of each interior angle, X.
Divide sum by number of sides, 8.
1080  8 = 135
8
In the figure, AF GD . What is the value of x.
HCG = DCE
Vertical Angles
50
HCG = 15
DCE = 15
K D
15
D = 50
Corresponding
Angles
K and D
Supplementary
Angles
K + D = 180
K + 50 = 180
K
= 130
8
In the figure, AF GD . What is the value of x.
50
The sum of the
angles in ∆CDE
is equal to 180
15
x + 15 + 130 = 180
x + 145 = 180
x = 35
130 D
9 In the figure, VW = WX = VX = XY = YZ = XZ.
If VZ = 12, what is the perimeter of the triangle
VWX?
VZ = 12
VX = 6
VW = 6
WX = 6
Perimeter VWX = 6 + 6 + 6 = 18
10 The perimeter of an isosceles triangle is 20 inches,
its base measures 8 inches. Find the length of each
of its equal sides in inches.
x = length of each equal side
Perimeter = 20 inches
x + x + 8 = 20
2x + 8 = 20
–8 –8
2x = 12
x = 6
x
x
8
11 If each of the equal sides of an isosceles triangle is 10,
and the base is 16, what is the area of the triangle?
Find height (h)
Use Pythagorean Theorem
a2 + b2 = c2
h2 + 82 = 102
h2 + 64 = 100
–64 –64
h2 = 36
h = 6
1
A  bh
2
10
10
h
8
16
Base (b) = 16
11 If each of the equal sides of an isosceles triangle is 10,
and the base is 16, what is the area of the triangle?
Find height (h)
h=6
1
1
A  bh   16  6
2
2
1
1 96
A   96  
2
2 1
96
A
= 48
2
1
A  bh
2
10
10
6
8
16
Base (b) = 16
12 In the figure, E is the midpoint of side CB of
rectangle ABCD, and x = 45°. If AB is 3
centimeters, what is the area of rectangle ABCD,
in square centimeters?
∆DCE is isosceles
CD  CE
x =45
45
3
3
CD  3 CE  3
45
3
Area of rectangle ABCD
Length  Width = 6  3 = 18
3
6
13
A.
B.
C.
D.
E.
If the area of a right triangle is 16, the length of
the legs could be
1
Find Area: A  bh
2
1
1
h
8 and 2 A   8  2   16 = 8
2
2
1
1
12 and 4 A   12  4   48 = 24
2
2
1
1
10 and 6 A   10  6   60 = 30
2
2
1
1
20 and 12 A   20  12   240 = 120
2
2
1
1
32 and 1 A   32  1   32 = 16
2
2
b
14 In the figure, right triangle ABC is contained
within right triangle AED. What is the ratio of
the area of AED to the area of ABC?
AC  BC ∆ABC is Isosceles
45
∆AED Big Triangle
45
A = 45 D = 90 E = 45
∆AED is Isosceles
ED  AD
ED  8
AD  8
45
8
14 In the figure, right triangle ABC is contained
within right triangle AED. What is the ratio of
the area of AED to the area of ABC?
Area of ∆ABC Small Triangle
1
1
1
A  bh   6  6   36 = 18
2
2
2
45
45
Area of ∆AED Big Triangle
1
1
1
A  bh   8  8   64 = 32
2
2
2
Area AED 32 16
Ratio 
 
Area ABC 18 9
45
8
15
The figure above shows a square region divided
into four rectangular regions, three of which have
areas 5x, 5x, and x2, respectively. If the area of
MNOP is 64, what is the area of square QROS?
x
5
Area of square QROS
Length  Width
5
5
x
x
= 5  5
= 25
x
5
16
In the figure, CDE is an equilateral triangle
and ABCE is a square with an area of 1.
What is the perimeter of polygon ABCDE?
ABCE is a square
with an area of 1
s2
Area =
1 = s2
1=s
Perimeter of ABCDE
1+1+1+1+1=5
1
1
1
1
1
1
17
One-third of the area of a square is 12 square inches.
What is the perimeter of the square, in inches?
1
A  12
3
1 
3 A   312 
3 
A = 36
A = s2
s2 = 36
s=6
6
6
6
6
Perimeter = 4(6)
Perimeter = 24
18
All the dimensions of a certain rectangular solid are
integers greater than 1. If the volume is 126 cubic
inches and the height is 6 inches, what is the
perimeter of the base?
V = Volume
V = lwh
126 = l  w  6
h6
w
Base
l
Perimeter of Base
2l + 2w
= 2(7) + 2(3)
126 l  w  6

=
14
+
6
6
6
21 = l  w (Base) = 20
l = 3 or 7
w = 3 or 7
19 A rectangular solid has a square base. The volume
is 360 cubic inches and the height is 10 inches.
What is the perimeter of the base?
V = Volume
h10
w
Base
l
V = lwh
360 = l  w  10
360 l·w·10
=
10
10
36 = l  w (Base)
l = 6 and w = 6
Perimeter of Base
4(s)
= 4(6)
= 24
20 Cube A has an edge of 2. Each edge of cube A is
increased by 50%, creating a second cube B.
What is the ratio of the volume of cube A to cube B?
Cube A
Cube B
2
2
V = side3
V=
23
3
=8
V = 33 = 27
2
50% of 2 = .50  2 = 1
50% increase = 2 + 1 = 3
V = side3
3
3
Volume Cube A
Ratio 
Volume Cube B
8

27
21 Cube A has an edge of 2. Each edge of cube A is
increased by 50%, creating a second cube B.
The surface area of cube B is how much greater than
the surface area of cube A?
SA = Surface Area
Cube A
Cube B
SA = 6s2
2
2
2
SA = 6s2
= 622
= 64
= 24
3
3
3
SA Cube B
54
–
–
SA Cube A
24
=
30
= 632
= 69
= 54
22 How many wooden toy cubes with a 3-inch edge
can fit in a rectangular container with dimensions
3 inches by 21 inches by 15 inches?
V = Volume
V = side3
3
V=
33
= 27
V = lwh
V = 3  21  15
V = 945
15
3
21
Find number of toy cubes in rectangular (large) container
Volume of
Volume
of

Large Container
One Toy Cube
945

27
=
35
23 If you assume that there is no wasted ice, how many
smaller rectangular block ice cubes, dimensions
234, can be cut from two large blocks of ice? The
size of each block of ice is shown below.
V = Volume
4
6
V=lwh
V = 4  10  6
V = 240
Ice Cube
V=lwh
V=234
V = 24
10
Find number of ice cubes in one large block of ice
Ice cubes in
Volume of
Volume
of

two large
Large Block of Ice
One Ice Cube
240

24
= 10
blocks of ice
2(10) = 20
If cube B has an edge three times that of cube A,
the volume of cube B is how many times the
volume of cube A?
Strategy: Substitute a number for each cube edge.
24
V = Volume
Cube A
1
Cube B
V = s3
V = 33
V = 27
s3
V=
V = 13
V=1
31= 3
Volume Cube B ÷ Volume Cube A = 27 ÷ 1 = 27
The surface areas of the rectangular prism are
given. If the lengths of the edges are integers,
what is the volume in cubic inches?
25
Strategy: Use trial and error with different combinations
of numbers to find the area of each face.
24 sq in.
42 sq in.
28 sq in.
28
?
1
28
Combination #1
28 = 1  28
42 = 28  ?
NO
No integer
factor for 42
25
The surface areas of the rectangular prism are
given. If the lengths of the edges are integers,
what is the volume in cubic inches?
Strategy: Use trial and error with different combinations
of numbers to find the area of each face.
24 sq in.
3
8
42 sq in.
Combination #1
28 = 1  28
42 = 28  ?
Combination #2
28 sq in.
14
3
2
14
28 = 2  14
42 = 14  3
24 = 3  8
NO
No integer
factor for 42
NO
Parallel line
segments not
equal
25
The surface areas of the rectangular prism are
given. If the lengths of the edges are integers,
what is the volume in cubic inches?
Strategy: Use trial and error with different combinations
of numbers to find the area of each face.
24 sq in.
6
4
42 sq in.
7
28 sq in.
7
Combination #3
28 = 4  7
42 = 7  6
24 = 6  4
YES
Volume = l · w · h
Volume = 4 · 6 · 7
6
Volume = 168 cubic inches
4