Plane geometry
• Relations and properties of shapes (triangles,
rectangles, parallelograms, trapezoids, and circles),
angles, parallel lines, and perpendicular lines.
• What happens when you move or change these
shapes?
–Translations, rotations, reflections
• Proofs
–Justification, logic.
• Three-dimensional geometry
• Measurements: perimeter, area, and volume.
Plane geometry
• Circles
Source: http://www.erikthered.com/tutor/act-facts-and-formulas.pdf
Plane geometry
• Lines in a plane
• What do we know about a and b in both of these cases?
Source: http://www.erikthered.com/tutor/act-facts-and-formulas.pdf
Plane geometry
•
Other shape areas and perimeters.
•
•
•
•
•
If an angle is greater than 90, it is obtuse.
If an angle is less than 90, it is acute.
If an angle is 90, it is a right angle.
TRIANGLE: SUM OF ALL ANGLES = 180
SQUARE AND RECTANGLE: SUM OF ALL ANGLES = 360
Source: http://www.erikthered.com/tutor/act-facts-and-formulas.pdf
Plane geometry
•
Right Triangles
 How do you find the length of a side in a right triangle? Pythagorean Theorem.
•
Other Triangles: Equilateral (all three sides are equal), Isosceles (two equal sides),
and Similar (corresponding angles are equal and sides are in proportion).
Source: http://www.erikthered.com/tutor/act-facts-and-formulas.pdf
Plane Geometry Circles
6
Plane Geometry Circles
What is the equation of
these circles?
(x-1)2 + y2 = 1
(x-3)2 + (y-1)2 = 4
7
Plane Geometry Terms
Congruent = equal lengths
Co-linear = on same line
abc = the angle of b in the triangle abc
Acute = less than 90 degrees
(A cute little angle)
Obtuse = greater than 90 degrees
8
Trigonometry
• Trigonometric functions for right triangles:
–SINE
–COSINE
–TANGENT
Source: http://www.erikthered.com/tutor/act-facts-and-formulas.pdf
Source: http://www.mathsisfun.com
trigonometry
Source: http://www.mathsisfun.com
Miscellaneous Topics – You May See
These On The ACT Math
Fundamental Counting Principles
3 shirts, 2 pairs of pants, 4 sweaters – how many days
with a different outfit?
(3)(2)(4) = 24 day of a unique combination
How many different and unique phone numbers of a 7
digit number?
(10)(10)(10)(10)(10)(10)(10) = 107
11
Miscellaneous Topics – Probabilities – Examples
Given: 5 red marbles are placed in a bag along with 6 blue
marbles and 9 white marbles:
Question: if three white marbles are removed, what is the
probability the next marble removed will be white?
• Originally, there were 9 white marbles out of 20; with 3
white marbles removed, there are 6 out of 17 remaining.
The probability the next marble removed is white = 6/17.
Question: if 4 blue marbles are added to the original amount,
what is the probability the first marble removed is NOT white?
• Now there are 24 marbles total with 15 non-white. The
probability that the first marble removed is not white is
15/24.
12
Practice Questions
10. A rectangle has an area of 32 square feet and a perimeter of 24 feet. What is the
shortest of the side lengths, in feet, of the rectangle?
F. 1
G. 2
H. 3
J. 4
K. 8
11. In triangle ABC, the sum of the measures of ∠A and ∠B is 47°. What is the measure of
∠C ?
A. 47° B. 86°
C. 94°
D. 133° E. 143°
17. Hot Shot Electronics is designing a packing box for its new line of Acoustical Odyssey
speakers. The box is a rectangular prism of length 45 centimeters, width 30 centimeters,
and volume 81,000 cubic centimeters. What is the height, in centimeters, of the box?
A. 75 B. 60
C. 48
D. 27
E. 18
18. Four points, A, B, C, and D, lie on a circle having a circumference of 15 units. B is 2 units
counterclockwise from A. C is 5 units clockwise from A. D is 7 units clockwise from A and 8
units counterclockwise from A. What is the order of the points, starting with A and going
clockwise around the circle?
F. A, B, C, D
G. A, B, D, C
H. A, C, B, D
J. A, C, D, B
K. A, D, C, B
Practice Questions
20. The length of a rectangle is 3 times the length of a smaller rectangle. The 2 rectangles have
the same width. The area of the smaller rectangle is A square units. The area of the larger
rectangle is kA square units. Which of the following is the value of k ?
F. 1/9
G. 1/3
H. 1
J. 3
K. 9
22. The dimensions of the right triangle shown below are given in feet. What is sin,θ ?
27. In right triangle ACE below, BD is parallel to AE, and BD is
perpendicular to EC at D. The length of AC is 20 feet, the length
of BD is 3 feet, and the length of CD is 4 feet. What is the
length, in feet, of AE?
A. 10
B. 12
C. 15
D. 16
E. 17
Practice Questions
31. For triangle FGH, shown below, which of the following is an expression for y in terms of x ?
#31
#37
37. The figure below consists of a square and 2 semicircles, with dimensions as shown. What is
the outside perimeter, in centimeters, of the figure?
A. 8 + 8π
B. 16 + 8π
C. 16 + 16π
D. 32 + 8π
E. 32 + 16π
38. In the figure below, points E and F are the midpoints of sides AD and BC of rectangle ABCD,
point G is the intersection of AF and BE, and point H is the intersection of CE and DF. The
interior of ABCD except for the interior of EGFH is shaded. What is the ratio of the area of EGFH
to the area of the shaded region?
F. 1:2
G. 1:3
H. 1:4
J. 1:6
K. Cannot be determined from the given information
Practice Questions
40. What is the surface area, in square inches, of an 8-inch cube?
F. 512
G. 384 H. 320 J. 256
K. 192
42. According to the measurements given in the figure below,
which of the following expressions gives the distance, in miles,
from the boat to the dock?
F. 30 tan 52°
G. 30 cos 52°
H. 30 sin 52°
J. 30/cos 52°
K. 30/sin 52°
Practice Questions
Use the following information to answer questions 44–46.
The figure below shows the design of a circular stained glass panel on display at Hopewell’s
Antique Shop. Seams separate the pieces of the panel. All red triangular pieces shown are
congruent and have a common vertex with each adjoining triangular piece. The 2 squares
shown are inscribed in the circle. The diameter of the panel is 2 feet.
44. The design of the stained-glass panel has how many
lines of symmetry in the plane of the panel?
F. 02
G. 04
H. 08
J. 16
K. Infinitely many
45. What is the area of the stained-glass panel,
to the nearest 0.1 square foot?
A. 3.1
B. 4.0
C. 6.2
D. 8.0
E. 12.6
46. Kaya wants to install a new circular stained-glass window in her living room. The design
of the window will be identical to that of the panel. The diameter of the new window will
be 75% longer than the diameter of the panel. The new window will be how many feet
in diameter?
F. 1.50 G. 2.50 H. 2.75 J. 3.50 K. 4.00
Practice Questions
Answers
10. J
11. D
17. B
18. J
20. J
22. G
27. B
31. E
37. B
38. G
40. G
42. F
44. H
45. A
47. C
48. H
56. J
57. E