Summit Geometry Worksheet
ACT Prep Series • Geometry
SUMMIT GEOMETRY
Form 72G • Page 1
Student Name
Date
Class Period
SECTION 1: PLANE GEOMETRY STRATEGY
For plane geometry, rely on core rules: sum of triangle interior angles is \(180^\circ\), vertical angles are equal, and parallel lines yield equal alternate interior angles. Mark up the figures directly!
01
Plane Geometry: Parallel Lines
In the figure below, lines \(l_1\) and \(l_2\) are parallel. Transversal line \(t\) intersects both. If the measure of \(\angle A\) is \((3x - 15)^\circ\) and the measure of \(\angle B\) is \((2x + 10)^\circ\), what is the value of \(x\)?
A B l₁ l₂ t
A
5
B
25
C
31
D
37
E
185
02
Plane Geometry: Circles & Area
A circle is inscribed inside a square with a side length of 12 cm, such that the circle touches all four sides of the square. What is the area, in square centimeters, of the region inside the square but outside the circle?
12 cm
F
\(144 - 144\pi\)
G
\(144 - 36\pi\)
H
\(36 - 9\pi\)
J
\(12 - 6\pi\)
K
\(144\pi - 144\)
03
Plane Geometry: Triangles & Similarity
In right triangle \(\triangle ABC\), a line segment \(DE\) is drawn parallel to base \(BC\), with \(D\) on \(AB\) and \(E\) on \(AC\). If \(AD = 4\) inches, \(DB = 8\) inches, and the area of \(\triangle ADE\) is 6 square inches, what is the area, in square inches, of the quadrilateral \(DBCE\)?
A B C D E
A
12
B
18
C
48
D
54
E
96
Workspace (Show Your Work)
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ACT Prep Series • Geometry
SUMMIT GEOMETRY
Form 72G • Page 2
SECTION 2: COORDINATE GEOMETRY STRATEGY
Equations of circles are written as \((x - h)^2 + (y - k)^2 = r^2\). For lines, use slope-intercept form \(y = mx + b\). Remember that perpendicular lines have negative reciprocal slopes!
04
Coordinate Geometry: Midpoint
On a coordinate grid, segment \(PQ\) has midpoint \(M(2, -5)\). If the coordinates of point \(P\) are \((-4, 3)\), what are the coordinates of point \(Q\)?
F
\((-1, -1)\)
G
\((-3, 4)\)
H
\((8, -13)\)
J
\((8, -7)\)
K
\((-10, 11)\)
05
Coordinate Geometry: Circle Equations
Which of the following represents the equation of a circle centered at point \((3, -4)\) on a standard coordinate plane that passes through the point \((7, -1)\)?
(3,-4) (7,-1)
A
\((x-3)^2+(y+4)^2=25\)
B
\((x+3)^2+(y-4)^2=25\)
C
\((x-3)^2+(y+4)^2=5\)
D
\((x-7)^2+(y+1)^2=25\)
E
\((x-3)^2-(y+4)^2=25\)
06
Coordinate Geometry: Perpendicular Lines
Line \(m\) has the equation \(2x - 5y = 10\). Which of the following defines the equation of line \(n\), which is perpendicular to line \(m\) and passes through the point \((4, 1)\)?
F
\(y = -\frac{5}{2}x + 11\)
G
\(y = \frac{2}{5}x - \frac{3}{5}\)
H
\(y = -\frac{2}{5}x + \frac{13}{5}\)
J
\(y = \frac{5}{2}x - 9\)
K
\(y = -\frac{5}{2}x - 9\)
Workspace (Show Your Work)
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ACT Prep Series • Geometry
SUMMIT GEOMETRY
Form 72G • Page 3
SECTION 3: SOLID GEOMETRY STRATEGY
To solve 3D space diagonals, apply the extended Pythagorean theorem: \(d^2 = l^2 + w^2 + h^2\). For cylinders, volume is \(V = \pi r^2 h\). Doubling a radius increases volume by a factor of 4!
07
Solid Geometry: Volume Scaling
A solid right circular cylinder has an initial volume of \(V\) cubic inches. If the radius of its circular base is doubled, and its height is cut in half, what is the volume of the new cylinder in terms of \(V\)?
A
\(\frac{1}{2}V\)
B
\(V\)
C
\(2V\)
D
\(4V\)
E
\(8V\)
08
Solid Geometry: Space Diagonals
An architect designs a custom storage box in the shape of a rectangular prism. The box is 3 feet wide, 4 feet long, and 12 feet tall. What is the length, in feet, of the diagonal line connecting the bottom-left corner of the base to the top-right corner of the opposite side?
3 ft 4 ft 12 ft
F
5
G
\(\sqrt{153}\)
H
13
J
19
K
25
09
Solid Geometry: Spheres & Volume
A spherical ball of modeling clay with a radius of 3 inches is completely reshaped into a single cube. If there is no waste of clay during reshaping, which of the following expressions represents the side length, in inches, of the resulting clay cube?
A
\(\sqrt[3]{36\pi}\)
B
\(\sqrt[3]{12\pi}\)
C
\(6\pi\)
D
\(36\pi\)
E
\(\sqrt{36\pi}\)
Workspace (Show Your Work)
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ACT Prep Series • Geometry
SUMMIT GEOMETRY
Form 72G • Page 4
SECTION 4: TRIGONOMETRY STRATEGY
Remember SOH-CAH-TOA. Recognize that \(\sin^2 \theta + \cos^2 \theta = 1\), and \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). For functions like \(y = A\sin(Bx)\), amplitude is \(|A|\) and period is \(\frac{2\pi}{B}\).
10
Trigonometry: SOH-CAH-TOA Application
A ladder 15 feet long rests against the side of a vertical building, with the bottom of the ladder on flat, level ground. If the ladder makes an angle of \(58^\circ\) with the ground, which of the following expressions represents the height, in feet, of the top of the ladder above the ground?
F
\(15 \sin 58^\circ\)
G
\(15 \cos 58^\circ\)
H
\(\frac{15}{\sin 58^\circ}\)
J
\(15 \tan 58^\circ\)
K
\(\frac{15}{\cos 58^\circ}\)
11
Trigonometry: Identities
For all values of \(\theta\), which of the following is equivalent to the expression \(\frac{1 - \cos^2 \theta}{\sin \theta \cos \theta}\)?
A
\(\cos \theta\)
B
\(\sin \theta\)
C
\(\tan \theta\)
D
\(\cot \theta\)
E
\(1\)
12
Trigonometry: Graphs & Periodicity
What is the period of the trigonometric function graph defined by the equation \(y = -4\sin(3\theta) + 2\)?
F
\(3\)
G
\(2\pi\)
H
\(\frac{2\pi}{3}\)
J
\(\frac{\pi}{3}\)
K
\(6\pi\)
Workspace (Show Your Work)
STOP! END OF PRACTICE SECTION. Page 4 of 4
Summit Geometry Answer Key
ACT Prep Series • Answer Key
SUMMIT GEOMETRY ANNOTATED KEY
Solutions • Page 1
This annotated key provides detailed mathematical derivations, common student misconceptions, and time-saving shortcuts for the ACT exam. Correct options are highlighted in green.
01
Plane Geometry: Parallel Lines
Lines \(l_1 \parallel l_2\). \(\angle A = (3x - 15)^\circ\), \(\angle B = (2x + 10)^\circ\). Find \(x\).
A (5)
B (25) ✓
C (31)
D (37)
E (185)
ACT Step-by-Step Breakdown:
- Geometric Relationship: Angles \(A\) and \(B\) are alternate exterior angles relative to parallel lines \(l_1\) and \(l_2\). Therefore, their measures are equal (\(\angle A = \angle B\)).
- Equation: Set \(3x - 15 = 2x + 10 \Rightarrow x - 15 = 10 \Rightarrow x = 25\).
💡 ACT Trap: Setting their sum to \(180^\circ\) (\(5x - 5 = 180 \Rightarrow x = 37\)) yields incorrect choice D.
02
Plane Geometry: Circles & Area
Area inside a square with side length 12 but outside an inscribed circle.
F (\(144-144\pi\))
G (\(144-36\pi\)) ✓
H (\(36-9\pi\))
J (\(12-6\pi\))
K (\(144\pi-144\))
ACT Step-by-Step Breakdown:
- Square Area: \(\text{Area}_{\text{square}} = 12^2 = 144\text{ cm}^2\).
- Circle Radius: Inscribed circle diameter equals side length \(12\text{ cm}\), so radius \(r = 6\text{ cm}\).
- Circle Area: \(\text{Area}_{\text{circle}} = \pi r^2 = \pi (6)^2 = 36\pi\text{ cm}^2\).
- Difference: \(\text{Area}_{\text{square}} - \text{Area}_{\text{circle}} = 144 - 36\pi\).
03
Plane Geometry: Triangles & Similarity
Right triangle \(\triangle ABC\) with \(DE \parallel BC\). \(AD=4\), \(DB=8\), and \(\text{Area}(\triangle ADE)=6\). Find \(\text{Area}(DBCE)\).
A (12)
B (18)
C (48) ✓
D (54)
E (96)
ACT Step-by-Step Breakdown:
- Scale Factor: Total side \(AB = 4 + 8 = 12\). Scale factor \(k = \frac{AD}{AB} = \frac{4}{12} = \frac{1}{3}\).
- Area Ratio: Area ratio is \(k^2 = (\frac{1}{3})^2 = \frac{1}{9}\). Thus, \(\text{Area}(\triangle ABC) = 9 \times 6 = 54\text{ sq in}\).
