Geometry Grit Slides Arkansas EOC Prep: Part 2
Geometry Grit
Right Triangles, Circle Properties, and 3D Modeling
The Blueprint
01
Trig Tactics
SOH-CAH-TOA and inverse functions for angles.
02
Circle Circuit
Arcs, chords, tangents, and angle relationships.
03
Volume Vault
3D figures, cross-sections, and density.
Trig Tactics
G.SRT.C.8
The Ratio Rule
Label sides relative to the Reference Angle \(\theta\) .
SOH \(\sin(\theta) = \frac{Opp}{Hyp}\)
CAH \(\cos(\theta) = \frac{Adj}{Hyp}\)
TOA \(\tan(\theta) = \frac{Opp}{Adj}\)
\(\theta\) Opposite Adjacent Hypotenuse
Circle Circuit
G.C.A.2
Tangent
Exactly 1 intersection point.
Secant
Intersecting at 2 points.
Chord
Endpoints on the circle.
Arc
Part of circumference.
Central Tangent Line
Volume Vault
G.GMD.A.3
The Pyramid Principle
Prism & Cylinder
V = Bh
Pyramid & Cone
V = \(\frac{1}{3}Bh\)
EOC Trap: Slant height (\(\ell\)) is for Surface Area; vertical height (\(h\)) is for Volume.
Sphere Volume
V = \(\frac{4}{3}\pi r^3\)
Cavalieri's Principle
If height and cross-sectional areas match, volumes match.
EOC Drill 1
A 24-foot ladder is leaning against a building. The base is 7 feet away.
Find angle \(x\) at the ground.
A) \(x = \sin^{-1}(\frac{7}{24})\)
B) \(x = \cos^{-1}(\frac{7}{24})\)
C) \(x = \tan^{-1}(\frac{7}{24})\)
D) \(x = \cos^{-1}(\frac{24}{7})\)
24 ft 7 ft x
EOC Drill 2
In Circle P, \(\angle APB\) is a central angle of \(110^\circ\). Point C is on the major arc.
Find measure of \(\angle ACB\).
A) 55°
B) 70°
C) 110°
D) 220°
P A B C \(110^\circ\)
Modeling Masterkey
Density = \(\frac{\text{Mass}}{\text{Volume}}\) OR Density = \(\frac{\text{Population}}{\text{Area}}\)
Mass = Density \(\times\) Volume
Standard Stakes Worksheet Standard Stakes
Geometry EOC Part 2 | Practice
Name:
Date:
1
Trigonometric Modeling
A surveyor stands 50 meters from a cell tower. Using a clinometer, they measure the angle of elevation to the top as \(38^\circ\). The surveyor's eye level is 1.6 meters above the ground.
Which calculation determines the total height of the cell tower, \(h\)?
A) \(h = 50 \sin(38^\circ) + 1.6\)
B) \(h = 50 \tan(38^\circ) + 1.6\)
C) \(h = \frac{50}{\tan(38^\circ)} + 1.6\)
D) \(h = \frac{50}{\cos(38^\circ)} + 1.6\)
38° 50m h
Problem 1.1: Show Your Work
A wheelchair ramp has a length of 14 feet and a vertical rise of 1 foot. Calculate the angle of inclination to the nearest tenth of a degree.
Angle: __________________
2
Circle Geometry
2.1 Segments \(AB\) and \(AD\) are tangent to Circle C. If \(\angle BAD = 54^\circ\), find the measure of major arc \(BD\).
C A B D 54°
2.2 Chords \(XY\) and \(WZ\) intersect at P. If \(XP=8, PY=3, WP=6\), find the length of \(PZ\).
P X Y Z W
3
Volume & Density
A storage tank is in the shape of a cylinder with a cone on top. Both have a radius of 4 meters. The cylindrical part is 10 meters tall, and the cone part is 3 meters tall.
Part A: Total Volume (Exact \(\pi\))
Part B: Mass (\(850\text{ kg/m}^3\))
r = 4 10m 3m
EOC Challenge
Describe the 2D cross-section formed by a plane intersecting a sphere. Explain how its area changes as the plane moves away from the center.
Structural Terms Organizer Arkansas Standards G.MG.A.1 - 3
Structural Terms
EOC Prep
"Geometry is a visual language. If you can't name the structure, you can't build the proof."
The Circle Library
1. Inscribed vs. Central Angles
Define the relationship between the angle and the arc it intercepts:
2. Circumscribed Center
Describe where the center is located relative to the triangle:
The 3D Vault
3. Cavalieri's Principle
Explain why oblique and right solids have the same volume:
4. Population Density
Write the formula for Density in terms of Pop and Area:
Visual Identification
Secant
Tangent
Chord
Sector
Rapid Recap
Radian Measure
Vertical Slice
Major Arc Grit Guide Teacher Key Grit Guide
Teacher Key & Facilitation
Arkansas EOC
Part 2
G.SRT.C.8 Right triangle trig & Pythagorean Theorem.
G.C.A.2 Inscribed angles, chords, and tangents.
G.GMD.A.3 3D volume modeling and density.
Standard Stakes Solutions
1
Cell Tower Height
Choice B
\(h = 50\tan(38^\circ) + 1.6\). Rationale: Tangent ratio is used for Opposite (tower part) and Adjacent (distance).
1.1
Wheelchair Ramp Angle
4.1°
\(\sin(x) = 1/14 \Rightarrow x \approx 4.1^\circ\). Reminder: Ramp length = Hypotenuse.
2.1
Circle Tangent Angle
234°
Minor arc = 126°. Major arc = \(360 - 126\).
3
Storage Tank Volume & Mass
A: 176π m³
B: ~469,982 kg
\(V_{cyl} (160\pi) + V_{cone} (16\pi) = 176\pi\). Mass = \(V \cdot 850\). Result: \(176 \cdot 3.1415 \cdot 850\).
Facilitation Secrets
Language Precision Stress that central angle measures equal arc measure . Avoid saying "angle is the arc."
Ramp Modeling In AR-EOC prompts, ramp "length" is the hypotenuse. "Base" or "horizontal" refers to the adjacent side.
Density Context Units like \(kg/m^3\) = Volume (G.MG.A.1). Units like \(people/mile^2\) = Area (G.MG.A.2).
