📐 Geometry — Topics
Master the core topics and techniques for working with geometry in MATHCOUNTS.
Triangle Properties
Angle Sum Property
Method: Sum of angles in any triangle is 180° Example: If two angles are 60° and 80°, the third angle is 180° - 60° - 80° = 40°
Pitfall: Forgetting that sum is always 180° Fix: Always remember that angles in triangle sum to 180°
Triangle Inequality
Method: Sum of any two sides must be greater than third side Example: Can sides 3, 4, 8 form a triangle? 3 + 4 = 7 < 8, so no
Pitfall: Not checking all three combinations Fix: Always check all three combinations of sides
Pythagorean Theorem
Method: In right triangle, a² + b² = c² where c is hypotenuse Example: If legs are 3 and 4, hypotenuse is √(3² + 4²) = √25 = 5
Pitfall: Forgetting which side is hypotenuse Fix: Hypotenuse is always the longest side, opposite the right angle
Special Right Triangles
30°-60°-90°: Sides in ratio 1 : √3 : 2 45°-45°-90°: Sides in ratio 1 : 1 : √2 3-4-5: Sides in ratio 3 : 4 : 5 5-12-13: Sides in ratio 5 : 12 : 13
Example: In 30°-60°-90° triangle, if short leg is 2, hypotenuse is 4, long leg is 2√3
Pitfall: Forgetting the ratios Fix: Memorize the common ratios for special triangles
Quadrilateral Properties
Parallelogram Properties
Opposite sides: Equal and parallel Opposite angles: Equal Diagonals: Bisect each other Area: Base × height
Example: In parallelogram ABCD, if AB = 5 and height to AB is 3, area = 5 × 3 = 15
Pitfall: Confusing with rectangle Fix: Parallelogram has opposite sides equal, rectangle has right angles
Rectangle Properties
All properties of parallelogram plus: Four right angles: All angles are 90° Diagonals: Equal length and bisect each other Area: Length × width
Example: Rectangle with length 6 and width 4 has area 24 and perimeter 20
Pitfall: Forgetting that diagonals are equal Fix: Remember that rectangle diagonals are equal length
Square Properties
All properties of rectangle plus: All sides equal: All four sides have same length Diagonals: Equal length, perpendicular, and bisect each other Area: Side²
Example: Square with side 5 has area 25 and perimeter 20
Pitfall: Confusing with rectangle Fix: Square has all sides equal, rectangle has opposite sides equal
Trapezoid Properties
One pair of parallel sides: Called bases Area: (1/2) × (base₁ + base₂) × height Isosceles trapezoid: Non-parallel sides equal, base angles equal
Example: Trapezoid with bases 6 and 10, height 4 has area (1/2) × (6 + 10) × 4 = 32
Pitfall: Forgetting to average the bases Fix: Always average the two bases before multiplying by height
Circle Properties
Basic Circle Formulas
Circumference: C = 2πr Area: A = πr² Diameter: d = 2r
Example: Circle with radius 5 has circumference 10π and area 25π
Pitfall: Confusing circumference and area formulas Fix: Circumference has 2π, area has π²
Chord Properties
Perpendicular from center: Bisects chord Equal chords: Equidistant from center Power of a point: AB × BC = DB × BE
Example: If chord is 8 and distance from center is 3, radius is √(4² + 3²) = 5
Pitfall: Forgetting to use right triangle Fix: Use right triangle formed by radius, half-chord, and distance from center
Tangent Properties
Perpendicular to radius: At point of tangency Equal lengths: From external point to circle Power of a point: (Tangent length)² = (Secant length) × (External part)
Example: If tangent is 6 and secant is 9, external part is 9 - 6 = 3
Pitfall: Forgetting that tangent is perpendicular to radius Fix: Always use right triangle formed by tangent and radius
Similarity and Congruence
Similarity Tests
AA: Two angles equal SAS: Two sides proportional and included angle equal SSS: All three sides proportional
Example: Triangles with sides 3, 4, 5 and 6, 8, 10 are similar (ratio 2:1)
Pitfall: Forgetting to check all conditions Fix: Always verify all required conditions for similarity
Congruence Tests
SSS: All three sides equal SAS: Two sides and included angle equal ASA: Two angles and included side equal AAS: Two angles and non-included side equal HL: Hypotenuse and leg equal (right triangles only)
Example: Triangles with sides 3, 4, 5 and 3, 4, 5 are congruent (SSS)
Pitfall: Using wrong test Fix: Always use the appropriate test for the given information
Scale Factor
Definition: Ratio of corresponding sides Area ratio: (Scale factor)² Volume ratio: (Scale factor)³
Example: If scale factor is 2, area ratio is 4, volume ratio is 8
Pitfall: Forgetting to square for area, cube for volume Fix: Always use correct power for the dimension
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
Example: Triangle with base 6 and height 4 has area (1/2) × 6 × 4 = 12
Pitfall: Forgetting the (1/2) for triangle Fix: Always include (1/2) for triangle area
Composite Figures
Method: Break into basic shapes, find area of each, add together Example: L-shaped figure = rectangle + triangle
Pitfall: Not considering all parts Fix: Always break into all basic shapes
Perimeter Formulas
Rectangle: P = 2(length + width) Square: P = 4 × side Triangle: P = sum of all sides Circle: C = 2πr
Example: Rectangle with length 6 and width 4 has perimeter 2(6 + 4) = 20
Pitfall: Forgetting to double for rectangle Fix: Always double the sum for rectangle perimeter
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
Example: Cylinder with radius 3 and height 5 has volume π × 3² × 5 = 45π
Pitfall: Forgetting the (1/3) for cone Fix: Always include (1/3) for cone volume
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
Example: Cube with side 4 has surface area 6 × 4² = 96
Pitfall: Forgetting to include all faces Fix: Always count all faces for surface area
Composite Solids
Method: Break into basic shapes, find volume of each, add together Example: L-shaped solid = rectangular prism + triangular prism
Pitfall: Not considering all parts Fix: Always break into all basic shapes
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
Pitfall: Forgetting to square the differences Fix: Always square both differences before adding
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)
Pitfall: Forgetting to divide by 2 Fix: Always divide both coordinates by 2
Slope Formula
Formula: m = (y₂ - y₁)/(x₂ - x₁) Example: Slope of line through (1, 2) and (4, 6) = (6-2)/(4-1) = 4/3
Pitfall: Forgetting to subtract in same order Fix: Always subtract in same order for both coordinates
Equation of a Line
Slope-intercept form: y = mx + b Point-slope form: y - y₁ = m(x - x₁) Standard form: Ax + By = C
Example: Line through (1, 2) with slope 3 has equation y - 2 = 3(x - 1)
Pitfall: Forgetting to distribute Fix: Always distribute the slope when using point-slope form
Transformations
Translations
Method: Add translation vector to each coordinate Example: Translate (2, 3) by (1, 4) → (2+1, 3+4) = (3, 7)
Pitfall: Forgetting to add to both coordinates Fix: Always add to both x and y coordinates
Reflections
Over x-axis: (x, y) → (x, -y) Over y-axis: (x, y) → (-x, y) Over y = x: (x, y) → (y, x)
Example: Reflect (2, 3) over x-axis → (2, -3)
Pitfall: Forgetting to change sign Fix: Always change the sign of the appropriate coordinate
Rotations
90° counterclockwise: (x, y) → (-y, x) 180°: (x, y) → (-x, -y) 270° counterclockwise: (x, y) → (y, -x)
Example: Rotate (2, 3) 90° counterclockwise → (-3, 2)
Pitfall: Forgetting to swap coordinates Fix: Always swap coordinates and change signs appropriately
Dilations
Method: Multiply each coordinate by scale factor Example: Dilate (2, 3) by factor 2 → (2×2, 3×2) = (4, 6)
Pitfall: Forgetting to multiply both coordinates Fix: Always multiply both x and y coordinates by scale factor
Common Mistakes
Formula Mistakes
Mistake: Using wrong formula for area Fix: Always identify the shape first, then use correct formula
Mistake: Forgetting the (1/2) for triangle area Fix: Always include (1/2) for triangle area
Mistake: Confusing circumference and area formulas Fix: Circumference has 2π, area has π²
Angle Mistakes
Mistake: Forgetting that angles in triangle sum to 180° Fix: Always remember that angles in triangle sum to 180°
Mistake: Confusing complementary and supplementary Fix: Complementary adds to 90°, supplementary adds to 180°
Coordinate Mistakes
Mistake: Forgetting to square differences in distance formula Fix: Always square both differences before adding
Mistake: Forgetting to divide by 2 in midpoint formula Fix: Always divide both coordinates by 2
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