4.2.4

Shortest Distance from a Point to a Line

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Shortest Distance from a Point to a Line

The shortest distance from a point to a straight line is always the perpendicular distance. Here's how to identify it:

Perpendicular Distance

Perpendicular Distance

  • The shortest distance from a point to a line is always the perpendicular line.
  • This means the line connecting the point to the original line must form exactly a 90∘90^\circangle.
Example 1

Example 1

  • Which path from point P to the line is the shortest?

    • Identify the perpendicular path (the one forming a 90° angle).

    • Compare the measurements. The perpendicular path is 4 cm, while the slanted path is 5 cm.

    • The perpendicular path is always the shortest distance.

Jump to other topics
1

Number

1.1

Place Value & Ordering

1.1.1Ordering Integers1.1.2Ordering Decimals1.1.3Equal Symbols1.1.4Place Value
1.2

Numerical Operations

1.2.1Adding Integers1.2.2Subtracting Integers1.2.3Multiplying Integers1.2.4Dividing Integers1.2.5Dividing with a Remainder1.2.6Adding & Subtracting Decimals1.2.7Multiplying Decimals1.2.8Dividing Decimals1.2.9Adding & Subtracting Negative Numbers1.2.10Multiplying & Dividing Negative Numbers1.2.11Multiplying by Powers of Ten1.2.12Dividing by Powers of Ten
1.3

Fractions

1.3.1Writing Fractions1.3.2Simplifying Fractions1.3.3Equivalent Fractions1.3.4Constructing Fractions1.3.5Ordering Fractions1.3.6Recurring Decimal Notation1.3.7Mixed Numbers & Improper Fractions
1.4

Numerical Operations (Fractions)

1.4.1Adding & Subtracting Fractions1.4.2Add & Subtract Mixed Numbers & Improper Fractions1.4.3Multiplying Fractions1.4.4Multiplying Mixed Numbers1.4.5Reciprocals1.4.6Dividing Fractions1.4.7Dividing Mixed Numbers
1.5

Roots & Powers

1.5.1Roots1.5.2Integer Powers1.5.3Estimating Powers & Roots1.5.4Laws of Indicies (Multiplication)
1.6

Structure of Calculations

1.6.1Order of Operations (BIDMAS)
1.7

Numerical Skills

1.7.1Using a Calculator - Powers, Roots & Fractions
1.8

Factors & Multiples

1.8.1Divisibility Rules1.8.2Finding Multiples1.8.3Finding Factors1.8.4Finding Prime Numbers
1.9

Systematic Listing

1.9.1Systematic Listing
1.10

Standard Form

1.10.1Reading Standard Form1.10.2Converting to Standard Form (Positive Powers)1.10.3Converting to Standard Form (Negative Powers)1.10.4Converting Form Standard Form (Positive Powers)1.10.5Converting From Standard Form (Negative Powers)1.10.6Adding & Subtracting Standard Form1.10.7Standard Form on a Calculator
1.11

Fractions, Decimals & Percentages

1.11.1Converting Fractions, Decimals & Percentages1.11.2Numbers as Percentages1.11.3Fractions of Amounts
1.12

Percentage Change

1.12.1Percentages of Amounts (Calculator)1.12.2Percentages of Amounts (No Calculator)1.12.3Percentage Change
1.13

Rounding & Estimation

1.13.1Rounding Integers1.13.2Estimating Calculations1.13.3Rounding Decimals1.13.4Rounding With Significant Figures
1.14

Bounds & Accuracy

1.14.1Finding Error Intervals
2

Algebra

2.1

Algebraic Notation

2.1.1Function Machines2.1.2Using Algebraic Notation
2.2

Substitution & Simplification

2.2.1Substituting into Expressions with Operations2.2.2Substituting into Algebraic Expressions2.2.3Substituting into Real-Life Expressions2.2.4Simplifying Expressions by Collecting Like Terms
2.3

Brackets

2.3.1Expanding Single Brackets2.3.2Expanding & Simplifying with Single Brackets2.3.3Expanding Double Brackets2.3.4Factorising Into One Bracket
2.4

Algebraic Fractions

2.4.1Operations with Algebraic Fractions
2.5

Rearranging

2.5.1Changing the Subject
2.6

Functions

2.6.1Finding the Input of a Function
2.7

Solving Linear Equations

2.7.1Solving Linear Equations (One Step)2.7.2Solving Linear Equations (Multi-Step)2.7.3Solving Linear Equations (Unknown Both Sides)2.7.4Solving Equations (Unknown in Denominator)2.7.5Forming & Solving Linear Equations
2.8

Inequalities

2.8.1Representing Inequalities on a Number Line2.8.2Solving Linear Inequalities2.8.3Linear Inequalities with Unknowns on Both Sides
2.9

Simultaneous Equations

2.9.1Solving Simultaneous Equations Graphically2.9.2Solving Simultaneous Equations by Elimination
2.10

Sequences

2.10.1Term-to-Term Rules (Patterns)2.10.2Term-to-Term Rules (Numerical)2.10.3Substituting into a Given Rule2.10.4Position-to-Term Rules2.10.5Finding the Nth Term (Linear Sequences)2.10.6Simple Geometric Progressions
2.11

Coordinates

2.11.1Midpoints of Line Segments
2.12

Linear Graphs

2.12.1Equations of Straight Lines2.12.2Finding Line Equation from Point & Gradient2.12.3Finding Line Equations from Two Points2.12.4Parallel Line Equations
2.13

Quadratic Graphs

2.13.1Interpreting Quadratics
3

Ratio, Proportion & Rates of Change

3.1

Unit Conversion

3.1.1Using Clocks3.1.2Calculating with Time3.1.3Using Timetables3.1.4Using Calendars3.1.5Converting Units of Length3.1.6Converting Units of Mass3.1.7Converting Units of Capacity3.1.8Metric Conversions3.1.9Converting Units of Area & Volume
3.2

Compound Units

3.2.1Estimating & Measuring3.2.2Calculating Speed3.2.3Best Buy & Unit Pricing3.2.4Density, Mass & Volume
3.3

Ratio Fundamentals

3.3.1Ratio Notation & Simplification3.3.2Ratios as Fractions and Linear Equations3.3.3Sharing in a Ratio3.3.4Proportion Problems
3.4

Advanced Ratio and Scale

3.4.1Map Scales & Real Distances
3.5

Proportion

3.5.1Currency Conversion3.5.2Direct Proportion3.5.3Inverse Proportion
3.6

Algebraic Proportion

3.6.1Direct Proportion3.6.2Inverse Proportion
4

Geometry & Measures

4.1

Area & Perimeter

4.1.1Perimeter of Simple Shapes4.1.2Perimeter of Compound Shapes4.1.3Area of Triangles4.1.4Area of Rectangles & Parallelograms4.1.5Area of Trapezia4.1.6Area of Compound Shapes4.1.7Area of Compound Shapes with Triangles
4.2

Properties of Shapes

4.2.1Naming & Classifying Triangles4.2.2Naming & Classifying Quadrilaterals4.2.3Properties of Lines4.2.4Shortest Distance from a Point to a Line4.2.5Properties of 3D Shapes
4.3

Angles

4.3.1Classification of Angles4.3.2Angles on a Line & Around a Point4.3.3Combining Angle Facts in Triangles4.3.4Angles on Parallel Lines4.3.5Interior & Exterior Angles in Polygons
4.4

Trigonometry

4.4.1Pythagoras' Theorem4.4.2Sides in Right-Angles Triangles4.4.3Angles in Right-Angled Triangles
4.5

Similarity & Congruence

4.5.1Congruent Shapes4.5.2Length Scale Factors in Similar Shapes
4.6

Circle Geometry

4.6.1Identifying Parts of a Circle4.6.2Calculating Circumference4.6.3Calculating Area of a Circle4.6.4Calculating Arc Lengths4.6.5Calculating Area of Sectors
4.7

3D Representations

4.7.1Nets of 3D Shapes
4.8

Surface Area

4.8.1Surface Area of Cubes & Cuboids4.8.2Surface Area of Prisms4.8.3Surface Area of Cylinders
4.9

Volume

4.9.1Volume of Prisms4.9.2Volume of Cylinders
4.10

Transformations

4.10.1Translation using Vectors4.10.2Rotations4.10.3Enlargements & Positive Scale Factors
4.11

Scale Drawings

4.11.1Scale Drawings
4.12

Bearings

4.12.1Bearings
5

Probability

5.1

Single Events

5.1.1Writing Probabilities as Fractions5.1.2Mutually Exclusive Events5.1.3Expected Results
5.2

Combined Events

5.2.1Reading Venn Diagrams5.2.2Venn Diagrams (Set Notation)5.2.3Independent & Mutually Exclusive Events
5.3

Experimental Probability

5.3.1Relative Frequency & Convergence
6

Statistics

6.1

Averages & Range

6.1.1Range6.1.2Median6.1.3Mode6.1.4Mean6.1.5Interpreting Two-Way Tables
6.2

Scatter Graphs

6.2.1Lines of Best Fit
6.3

Collecting & Presenting Data

6.3.1Types of Data

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Properties of Lines

Properties of 3D Shapes