Geometric Patterns
1. / Elvira is stacking boxes of crackers as part of a display in a grocery store. To build a 4-level pyramid, she used 30 boxes of crackers, as shown in the diagram below.

If she follows the same pattern, how many boxes will Elvira use to build a 9-level pyramid?
A. / 81
B. / 285
C. / 204
D. / 385

Write your response here:
(show your work)

2. / Look at the pattern below.

If each of the triangles are congruent right triangles, which generalization can be made from the last picture in the pattern?
A. / c2 = a2 - b2
B. / b2 = a2 +c2
C. / a2 = b2 +c2
D. / c2 = a2 + b2

Write your response here:
(show your work)

3. / Sum of Interior Angles
# of Sides / Sum of Interior Angles
3 / 180°
4 / 360°
5 / 540°
n-gon / ?
According to the pattern in the table above, which of the expressions below can be used to find the sum of the interior angles of an n-gon?
A. / 180° ÷ (n - 2)
B. / 180° ÷ n
C. / 180°(n - 2)
D. / 180°(n)

Write your response here:
(show your work)

4. / Different length measurements in three triangles are shown in the following figure. In each triangle, the line segment MN is parallel to line segment BC.

Which of the following equations is true?
A. /
B. /
C. /
D. /

Write your response here:
(show your work)

5. / The following three triangles represent a pattern.

If the pattern continues, which of the following is equal to the ratio of the side opposite of the 60° angle and the hypotenuse for the nth triangle?
A. /
B. /
C. /
D. /

Write your response here:
(show your work)

6. / Look at the pattern below.
8, 12, 16, 20, ...
Which number sentence can be used to determine n, the 10th number in the pattern?
A. / n = 4 × 10
B. / n = 4 × 10 - 4
C. / n = 4 × 10 + 4
D. / n = 4 ÷ 10

Write your response here:
(show your work)

7. / Line AB is tangent to circle C at point B, and line AD is tangent to circle C at point D as shown in the following illustration.

As n increases, the values of change. Using the following table, determine which of the following expressions gives the value of in terms of .
A. / 360° - 5n°
B. / 360° - n·n°
C. / 180° - n°
D. / 180° - 5n°

Write your response here:
(show your work)

8. / For each of the circles shown below, an angle is formed with two tangent lines. The measures of two intercepted arcs are shown for two circles.

Which expression can be used to represent the m3 ?
A. /
B. /
C. /
D. /

Write your response here:
(show your work)

9. / Which expression represents the total volume of the pictures shown if each cube has a side length of e?

A. / c2 + e3
B. / c · e3
C. / c2 · e3
D. / c2 · e

Write your response here:
(show your work)

10. / Three polygons and the sum of the interior angles of each is shown below.

Which of the following is the sum of the interior angles of a polygon with 11 sides?
A. / 360°
B. / 1,800°
C. / 1,620°
D. / 1,980°

Write your response here:
(show your work)

11. / The measures of two inscribed angles and the measures of their intercepted arcs are shown in the figures below.

Which expression can be used to represent the angle XYZ?
A. /
B. /
C. /
D. /

Write your response here:
(show your work)

12. / Sylvester's Pizzeria has a party room to accommodate pizza parties. They have rectangular tables that can be placed together end-to-end to sit large groups of people together. Some sample seating arrangements are shown below.

Which of the following expressions can be used to determine the number of people who can sit as a group if t tables are joined together?
A. / 4(t + 1)
B. / 6t
C. / 2(2t - 1)
D. / 2(2t + 1)

Write your response here:
(show your work)

13. / Use the pattern below to make conjectures regarding any prism.
Prisms
Prism / Base / # of
Edges - Base / # of
Vertices / # of Edges / # of Faces
/ Triangle / 3 / 6 / 9 / 5
/ Rectangle / 4 / 8 / 12 / 6
/ Pentagon / 5 / 10 / 15 / 7
If n is the number of edges in the base of a prism, which of the following expressions could be used to find the number of faces of any prism?
A. / n
B. / 3n
C. / n + 2
D. / 2n

Write your response here:
(show your work)

14. / Look at the pattern below.

If n equals the number of sides of a polygon, what generalization can be made regarding n?
A. / 360° ÷ n = measure of each interior angle of a regular polygon
B. / 360° × n = measure of each interior angle of a regular polygon
C. / 360° ÷ n = measure of each exterior angle of a regular polygon
D. / 360° × n = measure of each exterior angle of a regular polygon

Write your response here:
(show your work)

15. / C.J. stacked cups into the three arrangements shown below.

If he continues stacking cups according to this pattern, how many cups will be in the fifth arrangement?
A. / 15
B. / 11
C. / 28
D. / 21

Write your response here:
(show your work)

16. / The first three patterns of a sequence of dots are shown below.

If the pattern continues indefinitely, which of the following expressions can be used to determine the number of dots in the nth term of the sequence?
A. / n(n + 2)
B. / n(n + 1)
C. / n2 + 2
D. / 3n

Write your response here:
(show your work)

17. / Which expression represents the total surface area (including the top and bottom) of a tower of cubes, where each cube has a side length of e? (Do not include the faces between two stacked cubes.)

A. / e2(4c + 10)
B. / e2(4c + 2)
C. / e2(c + 2)
D. / 4c · e2

Write your response here:
(show your work)

18. / Look at the pattern below.
52, 104, 208, 416, ...
Which number sentence can be used to determine n, the 9th number in the pattern?
A. / n = 26 × 92
B. / n = (26 × 9) - 5
C. / n = 26 × 29
D. / n = 26 ÷ 9

Write your response here:
(show your work)

19. / The first three patterns of a sequence of dots are shown below.

If the pattern continues indefinitely, which of the following expressions can be used to determine the number of dots in the nth term of the sequence?
A. / n(n + 5)
B. / 3n
C. / n(n + 3)
D. / n2 + 3

Write your response here:
(show your work)

20. /

A. /
B. /
C. /
D. /

Write your response here:
(show your work)

21. / Which expression represents the total volume of the pictures shown if each cube has a side length of e?

A. / c · e3
B. / c3 · e
C. / c3 · e3
D. / c3 + e3

Write your response here:
(show your work)

22. / The first three patterns of a sequence of dots is shown below.

If the pattern continues indefinitely, which of the following expressions can be used to determine the number of dots in the nth term of the sequence?
A. / 3n
B. / n(n + 3)
C. / n(n + 1)
D. / n2 + 3

Write your response here:
(show your work)

23. / In the following triangles, the medians are shown.

Which expression can be used to represent the distance CG in the largest triangle shown?
A. /
B. /
C. /
D. /

Write your response here:
(show your work)

24. / The diagram below shows the arrangement of seats in the first three rows of an auditorium.

If this pattern continues throughout the auditorium, how many seats will be in the 15th row?
A. / 58
B. / 60
C. / 66
D. / 62

Write your response here:
(show your work)

25. / The first three patterns of a sequence of dots is shown below.

If the pattern continues indefinitely, which of the following expressions can be used to determine the number of dots in the nth term of the sequence?
A. / n(n + 1)
B. / n2 + 2
C. / n(n + 2)
D. / 3n

Write your response here:
(show your work)

26. / Makyla is stacking sugar cubes at a restaurant. She made the structure shown in the diagram below with nine sugar cubes.

If Makyla continues to stack sugar cubes in the same pattern, how many sugar cubes will she need to make a structure with 7 levels?
A. / 36
B. / 49
C. / 25
D. / 13

Write your response here:
(show your work)

27. / Gregory spilled a bag of 120 beans on the floor, and he and his brother are taking turns picking them up. A diagram showing the beans left after Gregory's turn is shown below.

A diagram showing the beans left after Gregory's brother's turn is shown below.

How many beans will be on the floor after Gregory's next turn?
A. / 0
B. / 15
C. / 13
D. / 20

Write your response here:
(show your work)

28. / Toby is playing a game where the players remove tiles from the game board. The game board before the game started is below.

The game board after each of the first three rounds is shown below.

Round 1 /
Round 2 /
Round 3
If this pattern continues, which of the following could be the game board after the fourth round?

W. /
X.

Y. /
Z.
A. / W
B. / X
C. / Y
D. / Z

Write your response here:
(show your work)

29. / The picture below shows a formation of cheerleaders.

If this pattern continues, how many levels will a formation with 10 cheerleaders have?
A. / 4
B. / 5
C. / 3
D. / 6

Write your response here:
(show your work)

30. / Henry is laying tile in his bathroom in the pattern below.

Henry expands the pattern to cover his entire bathroom floor. If he uses 289 tiles total, how many white tiles does he use?
A. / 200
B. / 128
C. / 161
D. / 72

Write your response here:
(show your work)

Answers

1. / B285
2. / Dc2 = a2 + b2
3. / C180°(n - 2)
4. / C
5. / C
6. / Cn = 4 × 10 + 4
7. / D180° - 5n°
8. / A
9. / Cc2 · e3
10. / C1,620°
11. / D
12. / D2(2t + 1)
13. / Cn + 2
14. / C360° ÷ n = measure of each exterior angle of a regular polygon
15. / D21
16. / An(n + 2)
17. / Ae2(4c + 10)
18. / Cn = 26 × 29
19. / Cn(n + 3)
20. / B
21. / Cc3 · e3
22. / Dn2 + 3
23. / A
24. / D62
25. / Bn2 + 2
26. / B49
27. / B15
28. / CY
29. / A4
30. / B128

Explanations

1. /
Starting from the top of the pyramid, the first level has 1 box, the second level has 4 boxes, the third level has 9 boxes, and the fourth level has 16 boxes.
Since the number of boxes in each level is the square of the level number, the fifth level will have 25 boxes, the sixth level will have 36 boxes, and so on.
Add together the number of boxes in each level to find the total number of boxes in a 9-level pyramid.
1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 = 285
2. /
The last picture in this pattern is a copy of an ancient illustration of the Pythagorean theorem.

c2 = / (a - b)2 + 2ab
c2 = / a2 - 2ab + b2 + 2ab
c2 = / a2 + b2
3. /
Start with the angle measurements of the triangle, the quadrilateral, and the pentagon.
180°(3 - 2) = 180°(1) = 180°
180°(4 - 2) = 180°(2) = 360°
180°(5 - 2) = 180°(3) = 540°
Therefore, the sum of the interior angles of an n-gon can be found by using the expression 180°(n - 2).