Quick Overview — What's on the Regents
The Geometry Regents tests your ability to: reason with proofs, use formulas and coordinate geometry, apply circle theorems, solve area/volume problems, handle trigonometry in right triangles, and do transformations/constructions. You must show work & reasons for proof-style questions.
Table of Contents (click to expand)
- Foundations & Vocabulary
- Triangles: congruence & similarity
- Quadrilaterals & polygons
- Circle theorems
- Coordinate geometry (distance, midpoint, slope, equations)
- Right triangle trig
- Area, perimeter & volume
- Transformations & symmetry
- Constructions & loci
- Proof strategies & sample proofs
- Common Regents question types & tips
- Practice problems & solutions
1. Foundations & Vocabulary — Must-know terms
- Point, line, plane: basic undefined terms
- Collinear — points on same line; coplanar — points on same plane
- Parallel (‖) and Perpendicular (⊥)
- Segment, ray, angle, vertex, interior/exterior
- Complementary (sum 90°), Supplementary (sum 180°)
- Linear pair — adjacent supplementary angles
- Vertical angles — equal
- Transversal with parallel lines: corresponding, alternate interior/exterior, consecutive interior — know angle relationships
Corresponding = equal
Alt. interior = equal
Consecutive interior = supplementary
Vertical angles = equal
2. Triangles — Congruence & Similarity (ALL the tests)
Congruence postulates
- SSS — side-side-side
- SAS — side-angle-side
- ASA — angle-side-angle
- AAS — angle-angle-side
- HL — hypotenuse-leg (right triangles)
Similarity
- AA — two pairs of equal angles imply similarity
- SAS similarity and SSS similarity (proportional sides)
- Scale factor k: to map triangle → triangle, multiply side lengths by k
- Proportions & cross-multiplication are your BFFs on Regents
Special triangles
- 30°-60°-90°: sides = x, x√3, 2x (short leg x opposite 30°)
- 45°-45°-90°: sides = x, x, x√2 (legs equal; hyp = leg·√2)
Triangle centers
| Center | Constructed by | Property |
|---|---|---|
| Centroid | Intersection of medians | 2:1 from vertex; center of mass |
| Circumcenter | Perpendicular bisectors | Equidistant from vertices; center of circ circle |
| Incenter | Angle bisectors | Equidistant from sides; center of incircle |
| Orthocenter | Altitudes | Varies (acute inside, right on vertex, obtuse outside) |
Common Regents tasks
- Prove two triangles congruent — choose correct postulate and state reasons
- Use similarity to find missing side lengths (proportions)
- Use triangle inequality when checking possible side lengths
3. Quadrilaterals & Polygons
Parallelogram properties
- Opposite sides parallel & equal
- Opposite angles equal
- Diagonals bisect each other
Rectangle
- All parallelogram properties + all angles 90°
Rhombus
- All sides equal; diagonals perpendicular & bisect angles
Square
- Rectangle + rhombus properties
Trapezoid
- At least one pair of parallel sides; midsegment = (b1+b2)/2
Regular polygons
- Interior angle = 180°(n-2)/n
- Exterior angle = 360°/n
4. Circles — central, inscribed, chords, tangents
Essential formulas
Circumference = 2πr = πd Area = πr²
Arc & angle relationships
- Central angle measure = measure of intercepted arc
- Inscribed angle = 1/2 intercepted arc
- Angle formed by two chords: average of arcs
- Angle formed by tangent & chord = 1/2 intercepted arc
Chord & tangent theorems
- Equal chords subtend equal arcs
- Perpendicular from center bisects chord
- Power of a point: (external secant)(whole secant) = (tangent)² (useful for chord/segment problems)
Sector and arc length
Arc length = (θ/360)·2πr Sector area = (θ/360)·πr²
5. Coordinate Geometry — distance, midpoint, slope, equation of a line
Formulas
Distance: d = √[(x₂-x₁)² + (y₂-y₁)²]
Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2)
Slope: m = (y₂-y₁)/(x₂-x₁)
Point-slope: y - y₁ = m(x - x₁) Slope-intercept: y = mx + b
Common Regents uses
- Show lines parallel (equal slopes) or perpendicular (product slopes = -1)
- Find equation of perpendicular bisector (midpoint + negative reciprocal slope)
- Use distance to prove congruence (SSS) in coordinate proofs
6. Trigonometry in Right Triangles
Definitions
sinθ = opposite/hypotenuse cosθ = adjacent/hypotenuse tanθ = opposite/adjacent
Regents tips
- Use calculator in DEGREE mode
- Make sure to identify which side is opposite/adjacent
- Use inverse trig to find angles: θ = sin⁻¹(opposite/hyp)
- When given a trig value, check for exact values with special triangles (30°,45°,60°)
Law of Sines & Cosines (non-right triangles)
Law of Sines: (sin A)/a = (sin B)/b = (sin C)/c
Law of Cosines: c² = a² + b² - 2ab cos C
7. Area, Perimeter & Volume
Common area formulas
| Figure | Formula |
|---|---|
| Triangle | Area = 1/2 · b · h |
| Rectangle | lw |
| Parallelogram | bh |
| Trapezoid | (1/2)(b₁ + b₂)h |
| Circle | πr² |
Volume formulas
| Solid | Formula |
|---|---|
| Rectangular prism | V = lwh |
| Cylinder | V = πr²h |
| Pyramid | V = (1/3)·Base Area·height |
| Cone | V = (1/3)·πr²h |
| Sphere | V = (4/3)·πr³ |
Cross-sections & nets
- When slicing solids, draw new shapes and use similarity ratios to scale areas/volumes
- Nets help visualize surface area — label dimensions carefully
8. Transformations & Symmetry
Types
- Translation — slide (add vector)
- Reflection — mirror across line (x-axis, y-axis, or y=x etc.)
- Rotation — rotate about a point (90°, 180°, 270° clockwise or ccw)
- Dilation — scale by factor k (center of dilation important)
Coordinate rules
| Reflect across y-axis | (x,y) → (-x,y) |
| Reflect across x-axis | (x,y) → (x,-y) |
| Rotate 90° cw | (x,y) → (y,-x) |
| Rotate 90° ccw | (x,y) → (-y,x) |
| Rotate 180° | (x,y) → (-x,-y) |
| Dilation with factor k | (x,y) → (kx, ky) [if center at origin] |
9. Constructions & Loci — Compass & Straightedge
Classic constructions you must know
- Perpendicular bisector of a segment
- Angle bisector
- Perpendicular from a point to a line
- Copying a segment / angle
- Constructing an equilateral triangle
Construction tips
- Use arcs with same radius when making bisectors
- Label intersection points (needed for proofs)
- Write steps clearly when asked on the Regents: e.g., "With center A and radius AB draw arc ..."
10. Proof Strategies & Sample Proofs
Proof tips
- Always state what's given and what to prove
- Draw a clean diagram, label all known equalities & angles
- Use theorems as steps — cite them (Vertical ∠s, parallel lines, congruence postulates, etc.)
- Use short statements & reasons format for clarity
Sample: Prove triangles congruent (SAS)
Given: In ΔABC and ΔDEF, AB = DE, ∠B = ∠E, BC = EF. Prove ΔABC ≅ ΔDEF.
Proof:
- AB = DE (given)
- ∠B = ∠E (given)
- BC = EF (given)
- Therefore ΔABC ≅ ΔDEF by SAS.
Sample: Coordinate proof (distance to show triangle is isosceles)
Given: A(0,0), B(4,0), C(2,√12). Show triangle ABC is isosceles.
Work: find AB = √[(4-0)² + (0-0)²] = 4. AC = √[(2-0)² + (√12-0)²] = √(4+12) = √16 = 4. Thus AB = AC → isosceles.
11. Common Regents Items & Exam Strategy
Exam strategies
- Label diagrams — always mark given lengths/angles
- Show work. Unexplained answers often lose points
- Use proper units & box final answers
- On multi-step problems, include intermediate steps (write proportions, cross-multiply)
- Manage time — do easier questions first; come back to long proofs
- For constructions, write each compass/straightedge step clearly
Acronyms & mnemonics
- SOAP — Slope: Opposite/Away? (not official — just remember slope = rise/run)
- SSS-SAS-ASA-AAS-HL — triangle congruence postulates
- AA — similarity easiest check for angles
- HIPPO — History DBQ helper (if you also study AP world 😉)
12. Practice Problems (with step-by-step answers)
1) In triangle ABC, AB = 10, BC = 6, AC = 8. Is triangle ABC right? (use converse of Pythagorean)
2) Find equation of line through (2,3) with slope -2.
3) Circle with center O has chord AB = 10 and distance from center to chord (perpendicular) = 6. Find radius.
4) A dilation with center (0,0) and factor 3 sends point (2,-1) → ?
5) Find area of trapezoid with bases 8 and 14 and height 5.
6) In triangle ABC, angle A = 30°, angle B = 70°. Find angle C.
7) Find area of triangle with vertices (0,0), (6,0), (6,8).
8) A right triangle has hypotenuse 13 and one leg 5. Find other leg.
9) Given parallelogram ABCD: A(0,0), B(5,0), D(1,3). Find coordinates of C.
10) Circle with center (2,1) radius 5. Does point (6,4) lie inside, on, or outside the circle?
13. Formulas Cheat Sheet (quick reference)
Pythagorean: a² + b² = c²
Area triangle = 1/2 bh
Area circle = πr²
Circumference = 2πr
Distance = √[(x₂-x₁)² + (y₂-y₁)²]
Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2)
Slope = (y₂-y₁)/(x₂-x₁)
Volume cylinder = πr²h
Volume cone = (1/3)πr²h
14. Radicals, Exact Values & Calculator Tips
Radicals
- Simplify radicals (factor perfect squares)
- Rationalize denominators if asked
Exact trig values
- sin(30°)=1/2, cos(30°)=√3/2, tan(30°)=1/√3
- sin(45°)=√2/2, cos(45°)=√2/2, tan(45°)=1
- sin(60°)=√3/2, cos(60°)=1/2, tan(60°)=√3
Calculator tips
- Use DEG for degree trig problems
- if inverse trig gives weird angle, check quadrant
- When rounding, match Regents instructions (usually 3 decimal places)