📐 Geometry — Reference
Essential concepts and definitions for working with geometry in MATHCOUNTS.
Basic Geometric Concepts
Points, Lines, and Planes
Point: A location in space with no size Line: A straight path that extends infinitely in both directions Ray: A part of a line that starts at a point and extends infinitely in one direction Line segment: A part of a line between two points Plane: A flat surface that extends infinitely in all directions
Angles
Angle: The figure formed by two rays sharing a common endpoint (vertex) Acute angle: Less than 90° Right angle: Exactly 90° Obtuse angle: Between 90° and 180° Straight angle: Exactly 180° Reflex angle: Between 180° and 360°
Angle Relationships
Complementary angles: Two angles whose sum is 90° Supplementary angles: Two angles whose sum is 180° Vertical angles: Opposite angles formed by intersecting lines (equal) Adjacent angles: Angles that share a common side and vertex
Triangles
Triangle Types
Equilateral triangle: All sides equal, all angles equal (60°) Isosceles triangle: Two sides equal, two angles equal Scalene triangle: All sides different, all angles different Right triangle: One angle is 90° Acute triangle: All angles less than 90° Obtuse triangle: One angle greater than 90°
Triangle Properties
Sum of angles: 180° Triangle inequality: Sum of any two sides > third side Pythagorean theorem: In right triangle, a² + b² = c² Area: A = (1/2) × base × height
Special Right Triangles
30°-60°-90° triangle: Sides in ratio 1 : √3 : 2 45°-45°-90° triangle: Sides in ratio 1 : 1 : √2 3-4-5 triangle: Sides in ratio 3 : 4 : 5 5-12-13 triangle: Sides in ratio 5 : 12 : 13
Quadrilaterals
Parallelograms
Definition: Quadrilateral with opposite sides parallel Properties: Opposite sides equal, opposite angles equal, diagonals bisect each other Area: A = base × height Perimeter: P = 2(a + b)
Rectangles
Definition: Parallelogram with four right angles Properties: All properties of parallelogram plus four right angles Area: A = length × width Perimeter: P = 2(length + width)
Squares
Definition: Rectangle with all sides equal Properties: All properties of rectangle plus all sides equal Area: A = side² Perimeter: P = 4 × side
Rhombuses
Definition: Parallelogram with all sides equal Properties: All properties of parallelogram plus all sides equal Area: A = (1/2) × diagonal₁ × diagonal₂ Perimeter: P = 4 × side
Trapezoids
Definition: Quadrilateral with exactly one pair of parallel sides Properties: One pair of parallel sides (bases) Area: A = (1/2) × (base₁ + base₂) × height Perimeter: P = sum of all sides
Circles
Basic Circle Properties
Center: Point equidistant from all points on the circle Radius: Distance from center to any point on the circle Diameter: Distance across the circle through the center (2 × radius) Circumference: Distance around the circle (2πr) Area: πr²
Circle Relationships
Chord: Line segment connecting two points on the circle Secant: Line that intersects the circle at two points Tangent: Line that touches the circle at exactly one point Arc: Part of the circumference between two points Sector: Region bounded by two radii and an arc Segment: Region bounded by a chord and an arc
Power of a Point
Theorem: If two chords intersect, the product of the segments of one chord equals the product of the segments of the other chord Formula: AB × BC = DB × BE
Similarity and Congruence
Similarity
Definition: Figures with same shape but different size Properties: Corresponding angles equal, corresponding sides proportional Scale factor: Ratio of corresponding sides Area ratio: (Scale factor)² Volume ratio: (Scale factor)³
Congruence
Definition: Figures with same shape and size Properties: Corresponding angles equal, corresponding sides equal Tests: SSS, SAS, ASA, AAS, HL (for right triangles)
Area and Perimeter
Basic Area Formulas
Rectangle: A = length × width Square: A = side² Triangle: A = (1/2) × base × height Parallelogram: A = base × height Trapezoid: A = (1/2) × (base₁ + base₂) × height Circle: A = πr²
Basic Perimeter Formulas
Rectangle: P = 2(length + width) Square: P = 4 × side Triangle: P = sum of all sides Circle: C = 2πr
Composite Figures
Method: Break into basic shapes, find area of each, add together Example: L-shaped figure = rectangle + triangle
Volume and Surface Area
Basic Volume Formulas
Rectangular prism: V = length × width × height Cube: V = side³ Cylinder: V = πr²h Cone: V = (1/3)πr²h Sphere: V = (4/3)πr³
Basic Surface Area Formulas
Rectangular prism: SA = 2(lw + lh + wh) Cube: SA = 6 × side² Cylinder: SA = 2πr² + 2πrh Cone: SA = πr² + πrl Sphere: SA = 4πr²
Coordinate Geometry
Distance Formula
Formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²] Example: Distance between (1, 2) and (4, 6) = √[(4-1)² + (6-2)²] = √[9 + 16] = 5
Midpoint Formula
Formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2) Example: Midpoint of (1, 2) and (4, 6) = ((1+4)/2, (2+6)/2) = (2.5, 4)
Slope Formula
Formula: m = (y₂ - y₁)/(x₂ - x₁) Example: Slope of line through (1, 2) and (4, 6) = (6-2)/(4-1) = 4/3
Transformations
Translations
Definition: Moving a figure without rotating or reflecting Notation: (x, y) → (x + a, y + b) Example: Translate (2, 3) by (1, 4) → (3, 7)
Reflections
Definition: Flipping a figure over a line Notation: Reflect over x-axis: (x, y) → (x, -y) Example: Reflect (2, 3) over x-axis → (2, -3)
Rotations
Definition: Turning a figure around a point Notation: Rotate 90° counterclockwise: (x, y) → (-y, x) Example: Rotate (2, 3) 90° counterclockwise → (-3, 2)
Dilations
Definition: Enlarging or shrinking a figure Notation: Dilate by factor k: (x, y) → (kx, ky) Example: Dilate (2, 3) by factor 2 → (4, 6)
Common Applications
Real-world Problems
Architecture: Use geometry in building design Engineering: Use geometry in structural design Art: Use geometry in artistic composition Sports: Use geometry in field design and equipment
Problem-solving Strategies
Draw diagrams: Visualize the problem Identify given information: List what you know Use appropriate formulas: Apply the right formula Check your answer: Verify the result makes sense