Lesson 4: GRE Quantitative Reasoning - Geometry Fundamentals & Coordinate Geometry 📐

Introduction

Welcome to Lesson 4! Now that you've mastered arithmetic, algebra, and proportional reasoning, it's time to explore the visual world of geometry. Approximately 15-20% of GRE Quantitative questions involve geometric concepts, making this a crucial area for your preparation.

Geometry questions test your ability to visualize shapes, understand spatial relationships, and apply formulas strategically. Unlike pure algebra, geometry requires you to combine multiple concepts—often mixing angle properties, area formulas, and algebraic reasoning in a single problem.

💡 GRE Tip: The GRE loves to combine geometry with algebra. You'll frequently see problems where you need to set up equations based on geometric relationships.

Core Concepts: Angles and Lines 📏

Angle Fundamentals

Angles are measured in degrees, and understanding angle relationships is foundational to GRE geometry.

Angle Classifications:
+------------------+------------------+
| Type             | Measure          |
+------------------+------------------+
| Acute            | 0° < θ < 90°     |
| Right            | θ = 90°          |
| Obtuse           | 90° < θ < 180°   |
| Straight         | θ = 180°         |
| Reflex           | 180° < θ < 360°  |
+------------------+------------------+

Complementary angles: Two angles that sum to 90° Supplementary angles: Two angles that sum to 180°

Parallel Lines and Transversals ⚡

When a transversal (a line crossing two other lines) intersects parallel lines, it creates special angle relationships:

     Line 1 (parallel)
        ↓
    1 2 | 3 4
   -----+-----  ← Transversal
    5 6 | 7 8
        ↓
     Line 2 (parallel)

• Corresponding angles are equal: ∠1 = ∠5, ∠2 = ∠6, ∠3 = ∠7, ∠4 = ∠8
• Alternate interior angles are equal: ∠3 = ∠6, ∠4 = ∠5
• Alternate exterior angles are equal: ∠1 = ∠8, ∠2 = ∠7
• Consecutive interior angles are supplementary: ∠3 + ∠5 = 180°, ∠4 + ∠6 = 180°

⚠️ Common Mistake: Assuming lines are parallel when they're not explicitly stated or marked. Always check for parallel line indicators (arrows or statements).

Core Concepts: Triangles 🔺

Triangle Fundamentals

The most important geometric shape on the GRE is the triangle. Every triangle has these properties:

1. Sum of interior angles = 180°
2. Triangle Inequality: The sum of any two sides must be greater than the third side
3. Exterior Angle Theorem: An exterior angle equals the sum of the two non-adjacent interior angles

Triangle Classifications by Sides:
+------------------+---------------------------+
| Type             | Properties                |
+------------------+---------------------------+
| Equilateral      | 3 equal sides             |
|                  | 3 angles = 60° each       |
+------------------+---------------------------+
| Isosceles        | 2 equal sides             |
|                  | 2 equal angles (base ∠s)  |
+------------------+---------------------------+
| Scalene          | No equal sides            |
|                  | No equal angles           |
+------------------+---------------------------+

Special Right Triangles 🌟

Memorise these ratios—they appear constantly on the GRE!

45-45-90 Triangle (Isosceles Right Triangle):

        |\
        | \
      a |  \ a√2
        |   \
        |____\
          a

Side ratio: 1 : 1 : √2

30-60-90 Triangle:

        |\
        | \
      2a|  \ (hypotenuse)
        |   \
        |____\
       60°  a
       
Height = a√3
Side ratio: 1 : √3 : 2

💡 Mnemonic: For 30-60-90, remember "1-2-3": shortest side = 1, hypotenuse = 2, middle side = √3

Triangle Area and Perimeter

Area formulas:

• Standard: A = ½ × base × height
• Heron's Formula (when you know all three sides): A = √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2

Perimeter: P = sum of all sides

Core Concepts: Quadrilaterals and Polygons 🔷

Quadrilateral Properties

A quadrilateral is any four-sided polygon. Interior angles sum to 360°.

Special Quadrilaterals:
+------------------+---------------------------+
| Type             | Properties                |
+------------------+---------------------------+
| Square           | 4 equal sides             |
|                  | 4 right angles            |
|                  | Area = s²                 |
+------------------+---------------------------+
| Rectangle        | Opposite sides equal      |
|                  | 4 right angles            |
|                  | Area = length × width     |
+------------------+---------------------------+
| Parallelogram    | Opposite sides equal &    |
|                  | parallel                  |
|                  | Area = base × height      |
+------------------+---------------------------+
| Trapezoid        | Exactly 1 pair parallel   |
|                  | Area = ½(b₁+b₂) × h       |
+------------------+---------------------------+
| Rhombus          | 4 equal sides             |
|                  | Opposite angles equal     |
|                  | Area = ½ × d₁ × d₂        |
+------------------+---------------------------+

General Polygon Formula

For any polygon with n sides:

• Sum of interior angles = (n - 2) × 180°
• Each interior angle of regular polygon = [(n - 2) × 180°] / n
• Sum of exterior angles = 360° (always!)

🤔 Did you know? The exterior angle sum is always 360° regardless of the number of sides. This makes it easier to find individual exterior angles in regular polygons: just divide 360° by the number of sides!

Core Concepts: Circles ⭕

Circle Fundamentals

A circle is the set of all points equidistant from a center point.

Key terminology:

• Radius (r): Distance from center to any point on the circle
• Diameter (d): Distance across the circle through the center; d = 2r
• Circumference (C): Distance around the circle; C = 2πr = πd
• Area (A): A = πr²

Circle Theorems 🎯

1. Central Angle Theorem: A central angle has the same measure (in degrees) as its intercepted arc
2. Inscribed Angle Theorem: An inscribed angle is half the measure of its intercepted arc
3. Angle in a Semicircle: Any angle inscribed in a semicircle is a right angle (90°)
4. Tangent-Radius Theorem: A tangent line is perpendicular to the radius at the point of tangency

Circle with inscribed angle:
        Arc = 80°
         ___
        /   \
       /  θ  \    θ = 40° (inscribed angle)
      /_______\   = ½ × arc measure

💡 GRE Strategy: When you see a triangle inscribed in a circle with one side as the diameter, you automatically know one angle is 90°!

Core Concepts: Coordinate Geometry 📊

The Coordinate Plane

The coordinate plane consists of two perpendicular number lines (x-axis and y-axis) intersecting at the origin (0, 0).

Distance Formula

The distance between two points (x₁, y₁) and (x₂, y₂) is:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

This comes directly from the Pythagorean theorem!

Midpoint Formula

The midpoint M between two points (x₁, y₁) and (x₂, y₂) is:

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

🧠 Memory tip: "Average the x's, average the y's!"

Slope Formula

The slope (m) of a line through points (x₁, y₁) and (x₂, y₂) represents the rate of change:

m = (y₂ - y₁)/(x₂ - x₁) = rise/run

Slope Properties:
+------------------+---------------------------+
| Relationship     | Slope Property            |
+------------------+---------------------------+
| Parallel lines   | Same slope (m₁ = m₂)      |
| Perpendicular    | Negative reciprocals      |
| lines            | (m₁ × m₂ = -1)            |
| Horizontal line  | Slope = 0                 |
| Vertical line    | Slope undefined           |
+------------------+---------------------------+

Line Equations

Slope-intercept form: y = mx + b

• m = slope
• b = y-intercept (where line crosses y-axis)

Point-slope form: y - y₁ = m(x - x₁)

• Useful when you know a point (x₁, y₁) and the slope m

Example 1: Angle Relationships 🔍

Problem: In the figure below, lines l and m are parallel. If angle a = 65°, what is the measure of angle b?

     Line l
        ↓
      a | 
   -----+-----  ← Transversal
        | b
        ↓
     Line m

Solution:

Angles a and b are consecutive interior angles (also called co-interior angles). When two parallel lines are cut by a transversal, consecutive interior angles are supplementary, meaning they sum to 180°.

a + b = 180° 65° + b = 180° b = 180° - 65° b = 115°

💡 Key Insight: Recognizing the angle relationship saves time. You don't need to find every angle—go straight to the relationship!

Example 2: Triangle Problem with Algebra 🔺

Problem: In triangle ABC, angle A is 20° more than twice angle B, and angle C is 10° less than angle B. What is the measure of angle A?

Solution:

Let angle B = x

Then:

• Angle A = 2x + 20
• Angle C = x - 10

Using the triangle angle sum property: A + B + C = 180° (2x + 20) + x + (x - 10) = 180 4x + 10 = 180 4x = 170 x = 42.5°

Therefore: Angle A = 2(42.5) + 20 = 85 + 20 = 105°

⚠️ Common Mistake: Stopping after finding x! The question asks for angle A, not angle B, so you must substitute back.

Example 3: Coordinate Geometry Application 📍

Problem: Points P(2, 5) and Q(8, 13) are endpoints of a line segment. What is the length of PQ?

Solution:

Using the distance formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²] d = √[(8 - 2)² + (13 - 5)²] d = √[6² + 8²] d = √[36 + 64] d = √100 d = 10

💡 Pattern Recognition: The differences (6 and 8) form a 3-4-5 Pythagorean triple scaled by 2! Recognizing common triples (3-4-5, 5-12-13, 8-15-17) can save calculation time.

Example 4: Circle and Triangle Combined 🔺⭕

Problem: A circle has center O and radius 5. Point A is on the circle, and point B is on the circle such that AB is a diameter. Point C is also on the circle. What is the area of triangle ABC if AC = 6 and BC = 8?

Solution:

Since AB is a diameter and C is on the circle, angle ACB is inscribed in a semicircle, which means angle ACB = 90° (angle in a semicircle theorem).

Now we have a right triangle with legs AC = 6 and BC = 8.

Area of triangle = ½ × base × height Area = ½ × 6 × 8 Area = 24

🧠 Strategic Thinking: This problem tests whether you recognize the semicircle property. Once you know it's a right triangle, the calculation becomes straightforward!

Also note: We can verify this is consistent since the hypotenuse AB should equal the diameter (2r = 10), and indeed √(6² + 8²) = √100 = 10. ✓

Common Mistakes to Avoid ⚠️

1. Assuming Figures Are Drawn to Scale

GRE figures are not necessarily drawn to scale unless explicitly stated. Don't rely on visual estimation!

2. Forgetting to Use Given Information

Every piece of information in a GRE problem has a purpose. If parallel lines are mentioned, use parallel line properties!

3. Confusing Radius and Diameter

This simple mix-up can double or halve your answer. Always identify which one you're given and which one you need.

4. Not Drawing Diagrams

For coordinate geometry or complex shape problems, sketch a quick diagram. Visual representation prevents errors.

5. Mixing Up Perpendicular and Parallel Slope Relationships

• Parallel lines: same slope (m₁ = m₂)
• Perpendicular lines: negative reciprocal slopes (m₁ = -1/m₂)

6. Forgetting to Simplify Radicals

The GRE often requires answers in simplified form. √50 should be written as 5√2.

7. Using the Wrong Formula

Double-check which shape you're working with:

• Triangle area: ½bh (not bh)
• Trapezoid area: ½(b₁ + b₂)h (not (b₁ + b₂)h)

Key Takeaways 🎯

✅ Master angle relationships: complementary (90°), supplementary (180°), vertical angles, and parallel line properties

✅ Memorize special right triangles: 45-45-90 (1:1:√2) and 30-60-90 (1:√3:2) ratios appear frequently

✅ Know polygon formulas: interior angle sum = (n-2)×180°; exterior angle sum always = 360°

✅ Circle essentials: C = 2πr, A = πr², and inscribed angle = ½ arc measure

✅ Coordinate geometry toolkit: distance formula, midpoint formula, and slope relationships (parallel = same; perpendicular = negative reciprocal)

✅ Combine concepts: GRE geometry questions often mix algebra with geometric relationships

✅ Draw diagrams: Visual representation helps identify relationships and prevents calculation errors

✅ Check for special cases: right triangles, perpendicular lines, angles in semicircles

Quick Reference Card 📋

╔══════════════════════════════════════════════════════════╗
║           GRE GEOMETRY QUICK REFERENCE                   ║
╠══════════════════════════════════════════════════════════╣
║ ANGLES                                                   ║
║ • Triangle sum: 180°                                     ║
║ • Quadrilateral sum: 360°                                ║
║ • Polygon sum: (n-2)×180°                                ║
║ • Complementary: sum = 90°                               ║
║ • Supplementary: sum = 180°                              ║
╠══════════════════════════════════════════════════════════╣
║ TRIANGLES                                                ║
║ • Area = ½bh                                             ║
║ • Pythagorean: a² + b² = c²                              ║
║ • 45-45-90 ratio: x : x : x√2                            ║
║ • 30-60-90 ratio: x : x√3 : 2x                           ║
╠══════════════════════════════════════════════════════════╣
║ CIRCLES                                                  ║
║ • Circumference: C = 2πr = πd                            ║
║ • Area: A = πr²                                          ║
║ • Inscribed angle = ½ arc                                ║
║ • Angle in semicircle = 90°                              ║
╠══════════════════════════════════════════════════════════╣
║ QUADRILATERALS                                           ║
║ • Rectangle: A = lw                                      ║
║ • Square: A = s²                                         ║
║ • Parallelogram: A = bh                                  ║
║ • Trapezoid: A = ½(b₁+b₂)h                               ║
╠══════════════════════════════════════════════════════════╣
║ COORDINATE GEOMETRY                                      ║
║ • Distance: d = √[(x₂-x₁)² + (y₂-y₁)²]                  ║
║ • Midpoint: M = ((x₁+x₂)/2, (y₁+y₂)/2)                  ║
║ • Slope: m = (y₂-y₁)/(x₂-x₁)                             ║
║ • Parallel: m₁ = m₂                                      ║
║ • Perpendicular: m₁×m₂ = -1                              ║
╚══════════════════════════════════════════════════════════╝

Further Study 📚

1. Khan Academy - GRE Geometry: https://www.khanacademy.org/test-prep/gre/gre-math - Comprehensive video lessons covering all geometry topics with practice problems

2. ETS Official GRE Quantitative Reasoning Practice: https://www.ets.org/gre/test-takers/general-test/prepare.html - Official practice questions from the test makers, including geometry problems with detailed explanations

3. Manhattan Prep GRE Geometry Strategy Guide: https://www.manhattanprep.com/gre/store/online-resources/ - In-depth strategies specifically for tackling GRE geometry questions efficiently

Congratulations on completing Lesson 4! You now have the geometric foundation to tackle visual reasoning problems on the GRE. In the next lesson, we'll explore data interpretation and statistics. Keep practicing! 🚀