Lecture Notes: Geometry Problem Solving

Introduction

• Focus on solving basic geometry problems
• Recommendation: Pause and solve problems yourself for effective learning

Problem 1: Complementary Angles

• Given: Angles ABD = 5x + 18, CBD = 7x, ABC = right angle
• Concept: ABD and CBD are complementary (add to 90°)
• Solution:
  • Equation: 5x + 18 + 7x = 90
  • Solve: 12x + 18 = 90 → x = 6
  • Angle ABD = 5(6) + 18 = 48°

Problem 2: Midpoint and Segment Length

• Given: C is midpoint, AD = 24, BC = 4
• Concept: AC = CD = 12
• Solution: AB = AC - BC = 12 - 4 = 8

Problem 3: Supplementary Angles and Quadratic Equation

• Given: Angles ABD = 10x + 20, CBD = x² - 11, ABC = straight line
• Concept: ABD + CBD = 180°
• Solution:
  • Equation: 10x + 20 + x² - 11 = 180
  • Factor: (x - 9)(x + 19) = 0, x = 9 (valid)
  • Angle CBD = 9² - 11 = 70°

Problem 4: Triangle Angle Ratios

• Given: Ratio 5:7:8, sum of angles = 180°
• Solution:
  • Equation: 5x + 7x + 8x = 180 → x = 9
  • Smallest angle = 5x = 45°, Largest = 8x = 72°
  • Difference = 72 - 45 = 27°

Problem 5: Circle, Radius, and Area of Shaded Region

• Given: Circumference = 20π, find area of shaded region
• Solution:
  • Radius r = 10
  • Area of circle = πr² = 100π
  • Area of triangle (right) = ½ * 10 * 10 = 50
  • Shaded area = 100π - 50

Problem 6: Equilateral Triangle Area

• Given: Side = 12
• Formula: Area = (√3/4) * s²
• Solution: Area = (√3/4) * 144 = 36√3

Problem 7: Parallel Lines and Transversals

• Concepts:
  • Alternate exterior angles are congruent
  • Corresponding angles are equal
• Solution:
  • Calculate x using angles in a triangle: 70 + 60 + x = 180 → x = 50

Problem 8: Diagonals in a Hexagon

• Formula: Diagonals = n(n-3)/2
• Solution: For n=6 (hexagon), Diagonals = 9

Problem 9: Exterior Angle of a Polygon

• Given: Regular pentagon
• Solution: Exterior angle = 360/n = 72°

Problem 10: Supplementary Angles in DMS

• Solution: 180° - 112°32'45'' = 67°27'15''

Problem 11: Circle Equation from Diameter Endpoints

• Steps:
  • Find center (midpoint formula)
  • Radius from midpoint to an endpoint
  • Standard circle equation

Problem 12: Area of Scalene Triangle

• Formula: Heron's formula
• Steps:
  • Calculate semi-perimeter
  • Use Heron’s formula for area

Problem 13: Rectangle Perimeter

• Given: Length:Width = 8:5, Area = 360
• Solution:
  • Solve for L and W
  • Calculate perimeter

Problem 14: Square’s Area from Diagonal

• Formula: Area = ½ * d²
• Solution: For diagonal d = 10, Area = 50*

Problem 15: Rhombus Area

• Formula: Area = ½ * d1 * d2
• Steps:
  • Use properties of rhombus to find diagonals
  • Calculate area

Problem 16: Kite Area

• Formula: Area = ½ * d1 * d2
• Steps:
  • Use properties of kite to find diagonals
  • Calculate area

Problem 17: Rectangular Prism Volume and Surface Area

• Given: Length = 12, Width = 5, Height = 4
• Solution:
  • Volume = L * W * H
  • Surface Area = 2(lw + wh + lh)

Problem 18: Similar Triangles

• Concept: Sides are proportional
• Solution:
  • Use proportions to find unknowns
  • Calculate angle sums

Problem 19: Midpoints and Segment Length

• Concept: Midpoints divide segments equally
• Solution: Use properties to solve for unknowns

Problem 20: Trapezoid Midsegment

• Concept: Midsegment is average of two bases
• Solution:
  • Calculate unknowns using midsegment properties

Problem 21: Quadrilateral Interior Angles

• Concept: Sum of interior angles = 360°
• Solution: Set up equation and solve for x

Problem 22: Slope of Perpendicular Lines

• Concept: Negative reciprocal of the slope
• Solution: Calculate slope of perpendicular line

Problem 23: Circle Chord and Tangent Relationships

• Concept: Use power theorems
• Solution:
  • Find length using secant-tangent power theorem

Problem 24: Cone Surface Area

• Given: Height = 15, Volume = 320π
• Solution:
  • Find radius using volume formula
  • Calculate total surface area

Problem 25: Arithmetic and Geometric Mean

• Concept: Mean formulas
• Solution:
  • Use arithmetic and geometric mean equations to solve

Problem 26: Altitude on Hypotenuse

• Concept: Geometric mean relationships
• Solution: Use known formulas to calculate lengths

Problem 27: Parallelogram Angle Measures

• Concept: Properties of parallelograms
• Solution: Use equal angle and side properties

Problem 28: Trigonometry and Right Triangles

• Concept: SOHCAHTOA for calculating sides
• Solution: Use tangent for side length

Problem 29: Regular Hexagon Area

• Formula: Area = ½ * apothem * perimeter
• Solution: Calculate apothem and perimeter

Problem 30: Triangular Prism Surface Area

• Formula: Total surface area = base area + lateral area
• Solution: Apply formula to find total surface area

Problem 31: Circle Arc Measure

• Concept: Tangent and secant angle relationships
• Solution: Calculate missing arc measures

Problem 32: Radii Sum in Circles

• Concept: Chord and tangent properties
• Solution: Calculate and sum radii

Conclusion

• Geometry problems often rely on understanding basic properties and relationships
• Use formulas and properties to solve problems efficiently

Study Tip: Focus on understanding the reasoning behind each formula and property in geometry. Practice with additional problems to strengthen your skills.