📐 Geometry — Problem Types

Master the common problem patterns and systematic solution approaches for geometry problems.

Triangle Problems

Angle Problems

Recognition: Questions asking for angle measures Template:

1. Identify given angles
2. Use angle sum property (180°)
3. Use angle relationships
4. Calculate unknown angle

Example: In triangle ABC, angle A = 60° and angle B = 80°. Find angle C.

1. Given: ∠A = 60°, ∠B = 80°
2. Property: Sum of angles = 180°
3. Calculation: ∠C = 180° - 60° - 80° = 40°
4. Answer: 40°

Common variations:

• Find missing angle in triangle
• Find angle in isosceles triangle
• Find angle in right triangle
• Find angle using exterior angle theorem

Side Length Problems

Recognition: Questions asking for side lengths Template:

1. Identify given information
2. Use appropriate theorem or formula
3. Calculate unknown side
4. Check answer

Example: In right triangle ABC, legs are 3 and 4. Find hypotenuse.

1. Given: Legs = 3, 4
2. Theorem: Pythagorean theorem
3. Calculation: c = √(3² + 4²) = √25 = 5
4. Answer: 5

Common variations:

• Find hypotenuse using Pythagorean theorem
• Find leg using Pythagorean theorem
• Find side using special right triangles
• Find side using similarity

Area Problems

Recognition: Questions asking for triangle area Template:

1. Identify base and height
2. Use area formula
3. Calculate area
4. Check answer

Example: Triangle with base 6 and height 4. Find area.

1. Given: Base = 6, height = 4
2. Formula: A = (1/2) × base × height
3. Calculation: A = (1/2) × 6 × 4 = 12
4. Answer: 12

Common variations:

• Find area using base and height
• Find area using Heron’s formula
• Find area using coordinates
• Find area using similarity

Quadrilateral Problems

Rectangle Problems

Recognition: Questions about rectangles Template:

1. Identify given information
2. Use rectangle properties
3. Apply appropriate formula
4. Calculate result

Example: Rectangle has length 6 and width 4. Find area and perimeter.

1. Given: Length = 6, width = 4
2. Properties: Opposite sides equal, four right angles
3. Formulas: A = l × w, P = 2(l + w)
4. Calculations: A = 6 × 4 = 24, P = 2(6 + 4) = 20
5. Answer: Area = 24, Perimeter = 20

Common variations:

• Find area and perimeter
• Find missing dimension
• Find diagonal length
• Find area using coordinates

Square Problems

Recognition: Questions about squares Template:

1. Identify given information
2. Use square properties
3. Apply appropriate formula
4. Calculate result

Example: Square has side 5. Find area and perimeter.

1. Given: Side = 5
2. Properties: All sides equal, four right angles
3. Formulas: A = s², P = 4s
4. Calculations: A = 5² = 25, P = 4 × 5 = 20
5. Answer: Area = 25, Perimeter = 20

Common variations:

• Find area and perimeter
• Find side length from area
• Find diagonal length
• Find area using coordinates

Parallelogram Problems

Recognition: Questions about parallelograms Template:

1. Identify given information
2. Use parallelogram properties
3. Apply appropriate formula
4. Calculate result

Example: Parallelogram has base 8 and height 5. Find area.

1. Given: Base = 8, height = 5
2. Properties: Opposite sides equal and parallel
3. Formula: A = base × height
4. Calculation: A = 8 × 5 = 40
5. Answer: 40

Common variations:

• Find area using base and height
• Find missing dimension
• Find angle measures
• Find area using coordinates

Trapezoid Problems

Recognition: Questions about trapezoids Template:

1. Identify given information
2. Use trapezoid properties
3. Apply appropriate formula
4. Calculate result

Example: Trapezoid has bases 6 and 10, height 4. Find area.

1. Given: Base₁ = 6, base₂ = 10, height = 4
2. Properties: One pair of parallel sides
3. Formula: A = (1/2) × (base₁ + base₂) × height
4. Calculation: A = (1/2) × (6 + 10) × 4 = 32
5. Answer: 32

Common variations:

• Find area using bases and height
• Find missing dimension
• Find angle measures
• Find area using coordinates

Circle Problems

Basic Circle Problems

Recognition: Questions about circles Template:

1. Identify given information
2. Use circle formulas
3. Calculate result
4. Check answer

Example: Circle has radius 5. Find circumference and area.

1. Given: Radius = 5
2. Formulas: C = 2πr, A = πr²
3. Calculations: C = 2π × 5 = 10π, A = π × 5² = 25π
4. Answer: Circumference = 10π, Area = 25π

Common variations:

• Find circumference and area
• Find radius from circumference
• Find radius from area
• Find diameter from radius

Chord Problems

Recognition: Questions about chords Template:

1. Identify given information
2. Use chord properties
3. Apply appropriate formula
4. Calculate result

Example: Chord is 8 units long and 3 units from center. Find radius.

1. Given: Chord length = 8, distance from center = 3
2. Property: Perpendicular from center bisects chord
3. Calculation: Half-chord = 4, radius = √(4² + 3²) = 5
4. Answer: 5

Common variations:

• Find radius using chord
• Find chord length using radius
• Find distance from center
• Find chord using power of a point

Tangent Problems

Recognition: Questions about tangents Template:

1. Identify given information
2. Use tangent properties
3. Apply appropriate formula
4. Calculate result

Example: Tangent is 6 units long and secant is 9 units long. Find external part.

1. Given: Tangent = 6, secant = 9
2. Property: Power of a point
3. Calculation: 6² = 9 × external part, so external part = 4
4. Answer: 4

Common variations:

• Find tangent length
• Find secant length
• Find external part
• Find radius using tangent

Similarity and Congruence Problems

Similarity Problems

Recognition: Questions about similar figures Template:

1. Identify similar figures
2. Find scale factor
3. Use similarity properties
4. Calculate result

Example: Two similar triangles have sides 3, 4, 5 and 6, 8, 10. Find scale factor.

1. Given: Sides 3, 4, 5 and 6, 8, 10
2. Scale factor: 6/3 = 2
3. Verification: 4 × 2 = 8, 5 × 2 = 10
4. Answer: 2

Common variations:

• Find scale factor
• Find missing side using similarity
• Find area ratio using similarity
• Find perimeter ratio using similarity

Congruence Problems

Recognition: Questions about congruent figures Template:

1. Identify congruent figures
2. Use congruence properties
3. Apply appropriate formula
4. Calculate result

Example: Two triangles have sides 3, 4, 5 and 3, 4, 5. Are they congruent?

1. Given: Sides 3, 4, 5 and 3, 4, 5
2. Test: SSS (all sides equal)
3. Conclusion: Yes, they are congruent
4. Answer: Yes

Common variations:

• Determine if figures are congruent
• Find missing side using congruence
• Find angle measure using congruence
• Find area using congruence

Area and Perimeter Problems

Basic Area Problems

Recognition: Questions asking for area Template:

1. Identify the shape
2. Use appropriate area formula
3. Calculate area
4. Check answer

Example: Rectangle has length 6 and width 4. Find area.

1. Shape: Rectangle
2. Formula: A = length × width
3. Calculation: A = 6 × 4 = 24
4. Answer: 24

Common variations:

• Find area of basic shapes
• Find area using coordinates
• Find area using similarity
• Find area using composite figures

Composite Area Problems

Recognition: Questions about composite figures Template:

1. Break into basic shapes
2. Find area of each shape
3. Add areas together
4. Check answer

Example: L-shaped figure with rectangle 6×4 and triangle with base 4 and height 3. Find total area.

1. Shapes: Rectangle and triangle
2. Areas: Rectangle = 6×4 = 24, Triangle = (1/2)×4×3 = 6
3. Total: 24 + 6 = 30
4. Answer: 30

Common variations:

• Find area of composite figures
• Find area using subtraction
• Find area using addition
• Find area using coordinates

Perimeter Problems

Recognition: Questions asking for perimeter Template:

1. Identify the shape
2. Use appropriate perimeter formula
3. Calculate perimeter
4. Check answer

Example: Rectangle has length 6 and width 4. Find perimeter.

1. Shape: Rectangle
2. Formula: P = 2(length + width)
3. Calculation: P = 2(6 + 4) = 20
4. Answer: 20

Common variations:

• Find perimeter of basic shapes
• Find perimeter using coordinates
• Find perimeter using similarity
• Find perimeter using composite figures

Volume and Surface Area Problems

Basic Volume Problems

Recognition: Questions asking for volume Template:

1. Identify the shape
2. Use appropriate volume formula
3. Calculate volume
4. Check answer

Example: Rectangular prism has length 6, width 4, height 3. Find volume.

1. Shape: Rectangular prism
2. Formula: V = length × width × height
3. Calculation: V = 6 × 4 × 3 = 72
4. Answer: 72

Common variations:

• Find volume of basic shapes
• Find volume using coordinates
• Find volume using similarity
• Find volume using composite solids

Surface Area Problems

Recognition: Questions asking for surface area Template:

1. Identify the shape
2. Use appropriate surface area formula
3. Calculate surface area
4. Check answer

Example: Cube has side 4. Find surface area.

1. Shape: Cube
2. Formula: SA = 6 × side²
3. Calculation: SA = 6 × 4² = 96
4. Answer: 96

Common variations:

• Find surface area of basic shapes
• Find surface area using coordinates
• Find surface area using similarity
• Find surface area using composite solids

Coordinate Geometry Problems

Distance Problems

Recognition: Questions asking for distance between points Template:

1. Identify the points
2. Use distance formula
3. Calculate distance
4. Check answer

Example: Find distance between (1, 2) and (4, 6).

1. Points: (1, 2) and (4, 6)
2. Formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²]
3. Calculation: d = √[(4-1)² + (6-2)²] = √[9 + 16] = 5
4. Answer: 5

Common variations:

• Find distance between two points
• Find distance from point to line
• Find distance using coordinates
• Find distance using similarity

Midpoint Problems

Recognition: Questions asking for midpoint Template:

1. Identify the points
2. Use midpoint formula
3. Calculate midpoint
4. Check answer

Example: Find midpoint of (1, 2) and (4, 6).

1. Points: (1, 2) and (4, 6)
2. Formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
3. Calculation: M = ((1+4)/2, (2+6)/2) = (2.5, 4)
4. Answer: (2.5, 4)

Common variations:

• Find midpoint of two points
• Find midpoint using coordinates
• Find midpoint using similarity
• Find midpoint using congruence

Slope Problems

Recognition: Questions asking for slope Template:

1. Identify the points
2. Use slope formula
3. Calculate slope
4. Check answer

Example: Find slope of line through (1, 2) and (4, 6).

1. Points: (1, 2) and (4, 6)
2. Formula: m = (y₂ - y₁)/(x₂ - x₁)
3. Calculation: m = (6-2)/(4-1) = 4/3
4. Answer: 4/3

Common variations:

• Find slope of line
• Find slope using coordinates
• Find slope using similarity
• Find slope using congruence

Common Mistakes and Fixes

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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