Congruence
27-34%of exam
Similarity, Right Triangles, and Trigonometry
29-37%of exam
Circles
2-8%of exam
Inscribed anglesCentral anglesTangentsChords/arcsSector area
Expressing Geometric Properties with Equations
12-18%of exam
Geometric Measurement and Dimensions
2-8%of exam
Modeling with Geometry
8-15%of exam
Scale drawingsDensityCapacity/costArea modelingOptimization
Quick Facts
- Exam
- Geometry Regents
- Credential
- NY Regents diploma
- Time
- 180 min
- Pass
- Scale 65
- Questions
- 35 total
- Fee
- School-administered
- Blueprint
- 6 domains
Congruence Criteria
SSS | SAS | ASA | AAS | HL
SSS: 3 sidesSAS: 2 sides + angleASA: 2 angles + sideAAS: 2 angles + sideHL: hyp + leg
Rigid Motion vs Dilation
Rigid motion
- Preserves size
- Translation/reflection/rotation
- Congruence
Dilation
- Changes size
- Scale factor
- Similarity
Preserves vs scales
Transformation Picker
- Slide fixed direction→Translation(Vector)
- Flip across line→Reflection(Mirror line)
- Turn about point→Rotation(Center+angle)
- Slide then flip→Glide reflection(Composition)
- Preserves size→Rigid motion(Congruence)
- Changes size→Dilation(Similarity)
- No motion→Identity(Same image)
Transformations
- Translation
- Slide by vector
- Reflection
- Flip over line
- Rotation
- Turn about point
- Glide reflection
- Translate then reflect
- Rigid motion
- Preserves size/shape
- Composition
- Sequence of motions
Triangle Congruence
- SSS
- 3 sides equalCongruent
- SAS
- 2 sides + included angle
- ASA
- 2 angles + included side
- AAS
- 2 angles + non-included side
- HL
- Hypotenuse + leg (right)
- CPCTC
- Parts congruent after proof
Constructions
- Copy segment
- Compass transfer
- Copy angle
- Compass arc method
- Perpendicular bisector
- Equidistant from endpoints
- Angle bisector
- Equal halves angle
- Parallel line
- Copy corresponding angle
SohCahToa
Soh | Cah | Toa
Soh: sin=opp/hypCah: cos=adj/hypToa: tan=opp/adj
Congruent vs Similar
Congruent
- Same size
- Same shape
- Rigid motion
Similar
- Same shape
- Proportional sides
- Dilation allowed
Equal vs proportional
Triangle Test Picker
- 3 sides equal→SSS(Congruent)
- 2 sides + included angle→SAS(Congruent)
- 2 angles + side→ASA/AAS(Congruent)
- Right + hyp + leg→HL(Congruent)
- 2 angles equal→AA(Similar)
- 3 proportional sides→SSS~(Similar)
- 2 proportional + angle→SAS~(Similar)
Similarity
- AA~
- 2 angles equalSimilar
- SSS~
- 3 proportional sides
- SAS~
- 2 proportional + included angle
- Dilation
- Scale about center
- Scale factor
- Ratio new/old
- Similarity ratio
- Side scale factor
Special Right Triangles
45-45-90 | 30-60-90
45-45-90: x | x | x√230-60-90: x | x√3 | 2x
Sine vs Cosine
Sine
- Opp/hyp
- Soh
- Height finder
Cosine
- Adj/hyp
- Cah
- Base finder
Opposite vs adjacent
Right Triangle Trig
- Sine
- Opp/hypSoh
- Cosine
- Adj/hypCah
- Tangent
- Opp/adjToa
- 45-45-90
- Legs equal; hyp = leg√2
- 30-60-90
- x | x√3 | 2x
- Geometric mean
- Altitude to hypotenuse
General Triangle Trig
- Law of Sines
- a/sinA = b/sinB
- Law of Cosines
- c² = a²+b²-2ab cosC
- Area formula
- ½ ab sinC
- Ambiguous case
- SSA may give 2 triangles
- Angle of elevation
- Horizontal up to line
Inscribed vs Central Angle
Inscribed
- Vertex on circle
- Half arc
- 2 endpoints
Central
- Vertex at center
- Equals arc
- 2 radii
On circle vs center
Circle Theorems
- Inscribed angle
- Half intercepted arc
- Central angle
- Equals intercepted arc
- Tangent
- Perp to radius
- Chord
- Endpoints on circle
- Secant
- Crosses 2 points
- Cyclic quadrilateral
- Opposite angles sum 180
Circle Measurements
- Arc length
- (θ/360) × 2πr
- Sector area
- (θ/360) × πr²
- Circumference
- 2πr or πd
- Chord length
- 2√(r²-d²)
- Tangent length
- From external point
Distance vs Midpoint
Distance
- Length between
- √(Δx²+Δy²)
- Positive scalar
Midpoint
- Center point
- Avg of coords
- A point
Length vs center
Coordinate Method Picker
- Prove parallel→Compare slopes(Equal)
- Prove perpendicular→Slope product = -1(Negative reciprocals)
- Find length→Distance formula(Δx² + Δy²)
- Find midpoint→Avg coordinates(Midpoint formula)
- Split segment→Partition ratio(Section formula)
- Circle center/radius→Complete the square((x-h)²+(y-k)²=r²)
Coordinate Tools
- Slope
- (y2-y1)/(x2-x1)
- Distance
- √(Δx²+Δy²)
- Midpoint
- Avg of coordinates
- Partition
- Section formula ratio
- Line equation
- y = mx + b
- Circle equation
- (x-h)²+(y-k)²=r²
Coordinate Proofs
- Parallel test
- Equal slopes
- Perpendicular test
- Slope product = -1
- Right angle test
- Perpendicular slopes
- Isosceles test
- Equal distances
- Midsegment
- Midpoints of 2 sides
Volume Formulas
Prism=Bh | Cylinder=πr²h | Cone=⅓πr²h
Prism: base×heightCylinder: πr²hCone: ⅓πr²hSphere: 4/3 πr³
Volume vs Surface Area
Volume
- 3D space
- Units³
- Prism = Bh
Surface area
- Total faces
- Units²
- Sum of areas
Inside vs outside
Volume and Solids
- Prism
- V = BhVolume
- Cylinder
- V = πr²h
- Cone
- V = ⅓πr²h
- Pyramid
- V = ⅓Bh
- Sphere
- V = 4/3 πr³
- Cross-section
- Slice of solid shape
- Solid of rotation
- Spin 2D figure
- Similar solids
- Vol ratio = k³
Modeling Contexts
- Scale drawing
- Ratio real/model
- Density
- Mass per volume
- Capacity
- Volume held
- Area cost
- Rate × area
- Optimization
- Max/min constraints
- Composite figure
- Sum/subtract parts
Common Traps
Scale Score vs Raw
65 is scaled ≠ Not percent correct
Required Tools
Graphing calculator ≠ Compass + straightedge
Proof Format
Parts II-IV want reasons ≠ Show every step
Similar vs Congruent
Similar = proportional ≠ Congruent = equal
Rigid vs Dilation
Rigid preserves size ≠ Dilation changes size
Conversion Chart
Each administration differs ≠ Use that chart only
Exact vs Approximate
Use exact π key ≠ Unless told to round
Last Minute
- 1.Big domains: congruence + similarity
- 2.35 Q | 80 credits | 3 hours
- 3.Pass = scale 65
- 4.SSS | SAS | ASA | AAS | HL
- 5.SohCahToa = sin/cos/tan
- 6.45-45-90 | 30-60-90
- 7.Volume: Prism=Bh | Cone=⅓πr²h
- 8.Distance + midpoint formulas
- 9.Inscribed = ½ arc
- 10.Graphing calc + compass required
- 11.Show reasons on proofs
- 12.Use exact π unless told
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