📐 Geometry Tactics

Master the essential geometric techniques that will help you solve problems involving shapes, angles, areas, and spatial relationships efficiently.

🎯 Core Geometric Strategies

Similar Triangles

• AA Similarity: Two angles equal
• SAS Similarity: Two sides proportional, included angle equal
• SSS Similarity: All three sides proportional
• Proportional relationships: Use similarity ratios

Power of a Point

• Secant-Secant: $PA \cdot PB = PC \cdot PD$
• Secant-Tangent: $PA \cdot PB = PT^2$
• Tangent-Tangent: $PT_1 = PT_2$
• Chord-Chord: $PA \cdot PB = PC \cdot PD$

Coordinate Geometry

• Distance formula: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
• Midpoint formula: $M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$
• Slope formula: $m = \frac{y_2-y_1}{x_2-x_1}$
• Area formulas: Triangle, rectangle, circle, etc.

🔍 Similar Triangles Mastery

AA Similarity

When to use: Two angles are equal in both triangles

Process:

1. Identify equal angles: Find corresponding equal angles
2. Set up proportion: Write ratio of corresponding sides
3. Solve for unknown: Use cross-multiplication
4. Check answer: Verify the solution

Example: In triangle ABC, DE || BC. If AD = 3, DB = 2, and DE = 6, find BC.

Solution:

• Since DE || BC, angles are equal (AA similarity)
• $\frac{AD}{AB} = \frac{DE}{BC}$
• $\frac{3}{3+2} = \frac{6}{BC}$
• $\frac{3}{5} = \frac{6}{BC}$, so $BC = 10$

SAS Similarity

When to use: Two sides proportional and included angle equal

Process:

1. Check side ratios: Verify corresponding sides are proportional
2. Check included angle: Verify the included angle is equal
3. Set up proportion: Write ratio of remaining sides
4. Solve for unknown: Use cross-multiplication

SSS Similarity

When to use: All three sides are proportional

Process:

1. Check all side ratios: Verify all corresponding sides are proportional
2. Set up proportions: Write ratios for any unknown sides
3. Solve for unknowns: Use cross-multiplication
4. Check answer: Verify the solution

⚡ Power of a Point Techniques

Secant-Secant Power

Formula: $PA \cdot PB = PC \cdot PD$

When to use: Two secants from external point P

Process:

1. Identify the point: Find the external point P
2. Identify secants: Find the two secants
3. Apply formula: $PA \cdot PB = PC \cdot PD$
4. Solve for unknown: Use the equation

Example: Point P is outside circle O. Secants PA and PC intersect the circle at A, B and C, D respectively. If PA = 8, PB = 4, and PC = 6, find PD.

Solution:

• By Power of a Point: $PA \cdot PB = PC \cdot PD$
• $8 \cdot 4 = 6 \cdot PD$
• $32 = 6 \cdot PD$, so $PD = \frac{32}{6} = \frac{16}{3}$

Secant-Tangent Power

Formula: $PA \cdot PB = PT^2$

When to use: One secant and one tangent from external point P

Process:

1. Identify the point: Find the external point P
2. Identify secant and tangent: Find the secant and tangent
3. Apply formula: $PA \cdot PB = PT^2$
4. Solve for unknown: Use the equation

Tangent-Tangent Power

Formula: $PT_1 = PT_2$

When to use: Two tangents from external point P

Process:

1. Identify the point: Find the external point P
2. Identify tangents: Find the two tangents
3. Apply formula: $PT_1 = PT_2$
4. Solve for unknown: Use the equation

📊 Coordinate Geometry Mastery

Distance Formula

Formula: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

When to use: Finding distance between two points

Process:

1. Identify points: Find coordinates of both points
2. Apply formula: Substitute into distance formula
3. Simplify: Calculate the distance
4. Check answer: Verify the result

Example: Find distance between (3, 4) and (7, 1).

Solution:

• $d = \sqrt{(7-3)^2 + (1-4)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

Midpoint Formula

Formula: $M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$

When to use: Finding midpoint of line segment

Process:

1. Identify endpoints: Find coordinates of both endpoints
2. Apply formula: Substitute into midpoint formula
3. Simplify: Calculate the midpoint
4. Check answer: Verify the result

Slope Formula

Formula: $m = \frac{y_2-y_1}{x_2-x_1}$

When to use: Finding slope of line through two points

Process:

1. Identify points: Find coordinates of both points
2. Apply formula: Substitute into slope formula
3. Simplify: Calculate the slope
4. Check answer: Verify the result

🎯 Advanced Geometric Techniques

Area Formulas

Triangle: $A = \frac{1}{2}bh$ or $A = \frac{1}{2}ab\sin C$ Rectangle: $A = lw$ Circle: $A = \pi r^2$ Trapezoid: $A = \frac{1}{2}h(b_1 + b_2)$ Parallelogram: $A = bh$

Shoelace Formula

For polygon with vertices $(x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)$: $$A = \frac{1}{2}\left|\sum_{i=1}^{n}(x_i y_{i+1} - x_{i+1} y_i)\right|$$ where $x_{n+1} = x_1$ and $y_{n+1} = y_1$

Shoelace Example

Problem: Find area of triangle with vertices (0,0), (3,0), (0,4).

Solution:

• Using shoelace formula: $A = \frac{1}{2}|(0 \cdot 0 + 3 \cdot 4 + 0 \cdot 0) - (0 \cdot 3 + 0 \cdot 0 + 4 \cdot 0)|$
• $A = \frac{1}{2}|(0 + 12 + 0) - (0 + 0 + 0)| = \frac{1}{2}|12| = 6$

Angle Properties

Triangle angles: Sum to 180° Quadrilateral angles: Sum to 360° Circle angles: Inscribed angle = half central angle Parallel lines: Corresponding angles equal, alternate angles equal Perpendicular lines: Right angles (90°)

⚡ Quick Geometric Tricks

Special Right Triangles

30-60-90: Sides in ratio $1 : \sqrt{3} : 2$ 45-45-90: Sides in ratio $1 : 1 : \sqrt{2}$ 3-4-5: Pythagorean triple 5-12-13: Pythagorean triple 8-15-17: Pythagorean triple

Circle Properties

Central angle: Angle with vertex at center Inscribed angle: Angle with vertex on circle Inscribed angle theorem: Inscribed angle = half central angle Thales’ theorem: Angle inscribed in semicircle is right angle Cyclic quadrilateral: Opposite angles sum to 180°

Area Ratios

Similar triangles: Area ratio = (side ratio)² Triangles with same height: Area ratio = base ratio Triangles with same base: Area ratio = height ratio Triangles sharing vertex: Area ratio = base ratio

🎯 Problem-Specific Strategies

Triangle Problems

• Use similarity: Look for similar triangles
• Apply area formulas: Use appropriate area formula
• Use coordinate geometry: Place triangle in coordinate system
• Apply angle properties: Use angle sum and relationships

Circle Problems

• Use power of a point: Apply power formulas
• Apply angle properties: Use inscribed and central angles
• Use coordinate geometry: Place circle in coordinate system
• Apply area formulas: Use circle area formula

Coordinate Problems

• Use distance formula: Find distances between points
• Use midpoint formula: Find midpoints
• Use slope formula: Find slopes
• Use area formulas: Find areas using coordinates

Angle Problems

• Use angle properties: Apply angle relationships
• Use parallel lines: Apply parallel line properties
• Use circle angles: Apply circle angle properties
• Use triangle angles: Apply triangle angle sum

🚨 Common Geometric Mistakes

Avoid These Errors:

• Similarity errors: Check that triangles are actually similar
• Power of a point errors: Apply the correct formula
• Coordinate errors: Check your coordinate calculations
• Area formula errors: Use the correct area formula
• Angle errors: Check angle relationships

Red Flags:

• Answer doesn’t make sense: Check your work
• Negative area: Check your calculations
• Impossible angle: Check your angle calculations
• Wrong coordinates: Check your coordinate work

📊 Quick Reference

Similarity Checklist:

• Identify equal angles: Find corresponding equal angles
• Set up proportion: Write ratio of corresponding sides
• Solve for unknown: Use cross-multiplication
• Check answer: Verify the solution

Power of a Point Checklist:

• Identify the point: Find the external point
• Identify secants/tangents: Find the relevant lines
• Apply formula: Use the appropriate power formula
• Solve for unknown: Use the equation

Coordinate Geometry Checklist:

• Identify points: Find coordinates of relevant points
• Apply formula: Use appropriate coordinate formula
• Simplify: Calculate the result
• Check answer: Verify the result

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