The concept that tangents drawn from an external point to a circle are equal in length is a fundamental principle in geometry. This theorem not only provides a foundation for understanding more complex geometric relationships but also plays a crucial role in various applications within the Cambridge IGCSE Mathematics curriculum (US - 0444 - Advanced). Mastery of this topic is essential for students aiming to excel in geometry by grasping the elegance and symmetry inherent in circular figures.

Key Concepts

Definition of a Tangent

A tangent to a circle is a straight line that touches the circle at exactly one point. This point is known as the point of tangency. Unlike a secant, which intersects the circle at two points, a tangent only grazes the circle, forming a right angle with the radius at the point of contact.

Properties of Tangents from an External Point

When two tangents are drawn from a single external point to a circle, several key properties emerge:

Equal Lengths: Both tangents are equal in length. If point P is external to circle O, and PT and PT' are the tangents from P to the circle, then PT = PT'.
Right Angle with Radius: Each tangent is perpendicular to the radius at the point of tangency. Therefore, ∠OTP and ∠OT'P are right angles.
Symmetry: The two radii (OT and OT') form an angle with equal angles on either side of the line connecting the external point P and the circle's center O.

Theorem Proof: Tangents from an External Point are Equal

To understand why tangents from an external point are equal in length, we can examine the geometric relationships involved. Consider circle O with center O. Let P be a point outside the circle, and PT and PT' be tangents from P to the circle, touching at points T and T' respectively. We aim to prove that PT = PT'.

Construct Radii: Draw radii OT and OT' to the points of tangency.
Right Angles: Since PT and PT' are tangents, OT ⊥ PT and OT' ⊥ PT'. Thus, ∠OTP and ∠OT'P are right angles.
Triangles Identical: Triangles OTP and OT'P are right-angled triangles sharing side OP.
Hypotenuse and Leg: In both triangles, OP is the hypotenuse, and OT and OT' are equal as radii of the circle.
By RHS Criterion: Right angle, Hypotenuse, and one Leg of both triangles are equal. Therefore, the triangles are congruent.
Conclusion: Corresponding sides PT and PT' are equal.

Mathematical Formula

The length of the tangent can be determined using the following relationship derived from the Pythagorean theorem: $$ PT = \sqrt{OP^2 - OT^2} $$ Where:

PT = Length of the tangent
OP = Distance from external point P to the center O
OT = Radius of the circle

Example Problem

Problem: Given a circle with center O and radius OT = 5 units. An external point P is located 13 units from O. Calculate the length of the tangent PT.

Solution: Using the formula: $$ PT = \sqrt{OP^2 - OT^2} = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12 \text{ units} $$ Thus, the length of the tangent PT is 12 units.

Applications of the Tangent Theorem

The theorem that tangents from an external point are equal in length has various applications:

Circle Geometry Problems: Solving for unknown lengths or angles in complex circular figures.
Construction and Engineering: Designing elements that require precise geometrical arrangements involving circles.
Optics: Understanding light paths where tangents represent instances of reflection.
Navigation: Determining paths that maintain a constant distance from a circular obstacle.

Related Theorems and Concepts

Several theorems and concepts are closely related to the tangents from an external point:

Power of a Point Theorem: Relates the lengths of tangents and secants from a common external point.
Alternate Segment Theorem: Connects angles in the alternate segment with tangents.
Chord Properties: Understanding chords in relation to tangents enhances problem-solving capabilities.

Diagrammatic Representation

A visual representation aids in comprehending the relationships: $$ \begin{align*} \text{Let O be the center of the circle, P the external point,} \\ \text{PT and PT' the tangents, and OT and OT' the radii.} \\ \end{align*} $$

Figure 1: Tangents PT and PT' from external point P to circle O.

Step-by-Step Proof Using Coordinates

Consider placing circle O at the origin of a coordinate system with radius r. Let P have coordinates (a, b) where the distance from P to O is greater than r. The equation of the circle is: $$ x^2 + y^2 = r^2 $$ The equation of a tangent to the circle at point (x₁, y₁) is: $$ xx₁ + yy₁ = r^2 $$ Solving the system of equations for two tangents yields points T and T' such that PT = PT', thereby proving the theorem algebraically.

Common Misconceptions

Understanding tangents involves addressing several common misconceptions:

Tangents Intersecting the Circle Twice: A line cannot be a tangent if it intersects the circle at two distinct points.
Assuming Unequal Tangents: From an external point, all tangents to a circle are equal, a property often overlooked.
Mistaking Secants for Tangents: Secants intersect the circle at two points, whereas tangents touch at only one.

Practice Exercises

Given a circle with radius 7 units and an external point 25 units from the center, find the length of the tangent from the external point to the circle.
Prove that if two tangents are drawn from an external point to a circle, the angles between each tangent and the line joining the external point to the center are equal.
A tangent to a circle is 9 units long from an external point. If the radius of the circle is 12 units, determine the distance from the external point to the center of the circle.

Advanced Concepts

Mathematical Derivation of Tangent Length

To derive the formula for the length of the tangent, consider the right-angled triangle formed by the radius, tangent, and the line from the external point to the center. Given:

O: Center of the circle
P: External point
PT: Tangent from P to point T on the circle
OT: Radius to the point of tangency

In triangle OPT:

OP: Hypotenuse
OT: One leg (radius)
PT: Other leg (tangent)

By the Pythagorean theorem: $$ OP^2 = OT^2 + PT^2 \\ PT^2 = OP^2 - OT^2 \\ PT = \sqrt{OP^2 - OT^2} $$ This derivation confirms the relationship between the lengths of OP, OT, and PT.

Exploring the Power of a Point

The Power of a Point theorem extends the concept of tangents and secants from an external point. It states that for a point P outside a circle, the product of the lengths of the two segments of any secant passing through P is equal to the square of the length of the tangent from P to the circle. Mathematically: $$ PT^2 = PA \cdot PB $$ Where:

PT = Length of tangent from P to the circle
PA and PB = Lengths of the secant segments from P

This theorem is instrumental in solving complex geometric problems involving multiple tangents and secants.

Application in Coordinate Geometry

Using coordinate geometry, the properties of tangents from an external point can be analyzed algebraically. For instance, calculating the exact points of tangency involves solving systems of equations representing the circle and the tangent lines. This approach reinforces the understanding of geometric properties through algebraic manipulation.

Intersections of Multiple Circles

When dealing with multiple circles, the tangents from an external point may intersect other circles in specific ways. Analyzing these scenarios involves understanding the relative positions and distances between circle centers and external points, employing the tangent length formula and the Power of a Point theorem.

Optimization Problems Involving Tangents

Optimization problems, such as finding the shortest or longest possible tangent under given constraints, require a deep understanding of tangent properties and their mathematical representations. Techniques from calculus and algebra are often employed to determine optimal solutions.

Integration with Trigonometry

Tangents intersect with trigonometric concepts when analyzing angles formed by tangents and radii. Trigonometric identities and ratios aid in solving problems involving angles, lengths, and distances related to tangents, enhancing the versatility of geometric problem-solving.

Real-World Engineering Applications

In engineering, the principles of tangents from an external point are applied in designing components that require precise curvature and connection points. This includes designing gears, cams, and other mechanical parts where circular motion and contact points are critical.

Advanced Proofs and Theorems

Beyond basic proofs, advanced theorems such as the Alternate Segment Theorem and properties of harmonic division involve tangents. These theorems require a higher level of logical reasoning and a deeper understanding of geometric principles.

Dynamic Geometry Software Applications

Utilizing dynamic geometry software like GeoGebra allows for interactive exploration of tangent properties. Students can manipulate points and observe real-time changes in tangent lengths and angles, fostering a more intuitive grasp of geometric relationships.

Case Study: Tangents in Architecture

Examining architectural designs reveals the application of tangent principles in creating aesthetically pleasing and structurally sound elements. Arches, domes, and other circular structures often incorporate tangent lines to ensure stability and visual harmony.

Complex Problem-Solving Strategies

Solving complex problems involving tangents requires multi-step reasoning and the integration of various geometric concepts. Strategies include breaking down problems into smaller parts, utilizing known theorems, and applying algebraic methods to find solutions.

Exploring Inscribed Angles

Inscribed angles related to tangents provide insights into the angular relationships within circles. Understanding how these angles interact with tangents enhances the ability to solve intricate geometric problems involving multiple circles and intersecting lines.

Advanced Construction Techniques

Constructing tangents using geometric tools demands precision and a thorough understanding of tangent properties. Advanced techniques involve leveraging compass and straightedge constructions to create accurate tangent lines from external points.

Mathematical Modelling with Tangents

Mathematical modelling involving tangents applies geometric principles to real-life scenarios. This includes modelling scenarios like optimizing paths, designing lenses in optics, and solving navigation problems where tangents play a crucial role.

Exploring Non-Euclidean Geometries

In non-Euclidean geometries, the concept of tangents takes on different characteristics. Exploring how tangents behave in spherical or hyperbolic geometries broadens the understanding of geometric principles beyond the Euclidean plane.

Comparison Table

Aspect	Tangents	Secants
Definition	A line that touches the circle at exactly one point.	A line that intersects the circle at two points.
Number of Intersection Points	One	Two
Perpendicular to Radius	Yes, at the point of tangency.	No, unless at special cases like diameters.
Length from External Point	Equal for tangents from the same external point.	Lengths can vary based on positions.
Applications	Geometry proofs, architectural designs, optics.	Chord lengths, arc measurements, intersection problems.

Summary and Key Takeaways

Tangents from an external point to a circle are equal in length.
Each tangent is perpendicular to the radius at the point of contact.
The Power of a Point theorem extends tangent properties to secants.
Understanding tangent properties is essential for advanced geometric problem-solving.
Applications of tangents span various fields, including engineering and optics.

Examiner Tip

Tips

Remember the acronym "TRP" – Tangent, Radius, Perpendicular – to recall that the tangent is perpendicular to the radius at the point of contact. Practice drawing precise diagrams to visualize problems better, and always double-check your calculations to avoid common pitfalls.

Did You Know

Did you know that the concept of tangents is not just limited to geometry? In computer graphics, tangents are crucial for rendering smooth curves and surfaces. Additionally, the tangent function in trigonometry shares its name due to its geometric origin related to tangent lines.

Common Mistakes

Students often confuse tangents with secants, mistakenly assuming that a tangent can intersect a circle at two points. Another common error is forgetting to apply the Pythagorean theorem correctly when calculating tangent lengths. Always ensure to verify the right angle between the tangent and the radius.

FAQ

What is a tangent in geometry?

A tangent is a straight line that touches a circle at exactly one point, known as the point of tangency.

Are all tangents from an external point equal?

Yes, all tangents drawn from the same external point to a circle are equal in length.

How do you prove that two tangents from an external point are equal?

By constructing radii to the points of tangency and applying the Right Hypotenuse Side (RHS) criterion, we can prove that the two tangent segments are congruent.

What is the relationship between tangents and the Power of a Point theorem?

The Power of a Point theorem relates the lengths of tangents and secants from a common external point, establishing that the square of the tangent length equals the product of the secant segments.

Can a tangent ever be a secant?

No, a tangent and a secant are distinct. A tangent touches the circle at one point, whereas a secant intersects the circle at two points.

How are tangents used in real-world applications?

Tangents are used in various fields such as engineering design, optics, navigation, and computer graphics to create precise curves and maintain specific distances.

1. Trigonometry

1.1 Right-Angled Triangles

1.1.1 Apply the Pythagorean Theorem to find missing sides

1.1.2 Use trigonometric ratios (sine, cosine, tangent) to solve right-angled triangles

1.2 Trigonometric Ratios

1.2.1 Know exact values for sine, cosine, and tangent of 0°, 30°, 45°, 60°, 90°

1.3 Bearings and Angles

1.3.1 Solve problems involving bearings

1.3.2 Understand and use the concepts of angle of elevation and angle of depression

1.4 Extended Trigonometry

1.4.1 Extend sine and cosine values to angles between 0° and 360°

1.4.2 Use the relationship between sine and cosine of complementary angles

1.5 Trigonometric Graphs

1.5.1 Graph and understand the properties of sine, cosine, and tangent functions

1.6 Sine Rule

1.6.1 Apply the Sine Rule for solving triangles (ASA, SSA cases)

1.7 Cosine Rule

1.7.1 Apply the Cosine Rule for solving triangles (SAS, SSS cases)

1.8 Triangle Area

1.8.1 Use the formula for the area of a triangle (1/2 * ab * sinC)

2. Transformations and Vectors

2.1 Transformations on Cartesian Plane

2.1.1 Describe and perform stretches

2.1.2 Describe and perform translations using column vectors

2.1.3 Describe and perform reflections in a given axis or line

2.1.4 Describe and perform rotations about a point

2.1.5 Describe and perform enlargements (dilations) with scale factors

2.2 Inverse Transformations

2.2.1 Find the inverse of a given transformation

2.3 Combined Transformations

2.3.1 Find the single transformation equivalent to a sequence of transformations

2.4 Vector Notation

2.4.1 Understand directed line segment notation and component form of vectors

2.4.2 Use appropriate symbols for vectors and their magnitudes

2.5 Vector Operations

2.5.1 Find the components of a vector by subtracting coordinates of an initial point from a terminal point

2.5.2 Use position vectors

2.5.3 Calculate the magnitude of a vector

2.5.4 Understand vector subtraction as v - w = v + (-w)

2.5.5 Multiply a vector by a scalar

2.5.6 Add and subtract vectors algebraically (component form) and geometrically (parallelogram rule)

3. Statistics

3.1 Line of Best Fit

3.1.1 Draw a straight line of best fit by eye through the mean on a scatter diagram

3.2 Data Interpretation

3.2.1 Read and interpret data from graphs and tables

3.3 Types of Data

3.3.1 Understand discrete and continuous data

3.4 Measures of Central Tendency

3.4.1 Calculate mean, mode, median, and range from discrete data

3.4.2 Calculate mean, modal class, median, and range from grouped and continuous data

3.5 Histograms

3.5.1 Construct and interpret histograms with frequency density on the vertical axis

3.5.2 Interpret histograms with unequal class intervals

3.6 Cumulative Frequency

3.6.1 Create and interpret cumulative frequency tables and curves

3.6.2 Determine median, quartiles, percentiles, and interquartile range

3.7 Comparing Data Sets

3.7.1 Use statistics (median, mean, interquartile range) to compare different data sets

3.8 Correlation

3.8.1 Understand and describe correlation (positive, negative, or zero) using scatter diagrams

3.9 Graphical Representation

3.9.1 Construct and interpret compound bar charts, dot plots, line graphs, pie charts, scatter diagrams

4. Geometry

4.1 Angles in Polygons

4.1.1 Angle properties of triangles, quadrilaterals, and polygons

4.1.2 Interior and exterior angles of a polygon

4.2 Geometrical Constructions

4.2.1 Make formal geometric constructions using a compass and straight edge

4.2.2 Copy and bisect a segment or an angle

4.2.3 Construct perpendicular lines and perpendicular bisectors

4.2.4 Construct equilateral triangles, squares, and hexagons inscribed in circles

4.2.5 Construct inscribed and circumscribed circles of a triangle

4.2.6 Construct tangent lines from a point outside a circle

4.3 Circles

4.3.1 Tangent perpendicular to radius at the point of contact

4.3.2 Tangents from a point

4.3.3 Angle in a semicircle

4.3.4 Angles at the center and at the circumference on the same arc

4.3.5 Cyclic quadrilateral properties

4.3.6 Equal chords are equidistant from the center

4.3.7 The perpendicular bisector of a chord passes through the center

4.3.8 Tangents from an external point are equal in length

4.4.2 Use area and volume scale factors for similar figures and solids

4.5 Congruence

4.5.1 Recognize and use congruence in solving geometric problems

4.6 Geometrical Vocabulary

4.6.1 Know definitions of acute, obtuse, right angle, reflex, equilateral, isosceles, congruent, similar,

4.6.2 Know definitions of pentagon, hexagon, octagon, rectangle, square, kite, rhombus, parallelogram, tra

4.7 Definitions

4.7.1 Understand definitions of angle, circle, perpendicular line, parallel line, and line segment

4.8 Symmetry

4.8.1 Line and rotational symmetry in 2D and 3D

4.8.2 Recognize symmetry properties of prisms and pyramids

4.9 Angles and Parallel Lines

4.9.1 Angles around a point

4.9.2 Angles on a straight line and intersecting straight lines

4.9.3 Vertically opposite angles

4.9.4 Alternate and corresponding angles on parallel lines

5. Functions

5.1 Function Notation

5.1.1 Use function notation

5.1.2 Understand domain and range

5.1.3 Use mapping diagrams

5.2 Graphs of Equations

5.2.1 Graph an equation in two variables as the set of all its solutions

5.2.2 Construct tables of values and graphs for functions of the form ax^n where n = -2, -1, 0, 1, 2, 3

5.2.3 Solve equations approximately using graphical methods

5.3 Writing Functions

5.3.1 Write a function that describes a relationship between two quantities

5.4 Comparing Functions

5.4.1 Compare properties of two functions represented in different ways (algebraically, graphically, numer

5.5 Recognizing Functions

5.5.1 Recognize linear, quadratic, cubic, reciprocal, exponential, and trigonometric functions from their

5.5.2 Interpret key features of function graphs (intercepts, increasing/decreasing behavior, maxima/minima

5.6 Domain and Range

5.6.1 Relate the domain of a function to its graph and quantitative relationships

5.7 Rate of Change

5.7.1 Calculate and interpret the average rate of change of a function over a specified interval

5.7.2 Estimate rate of change using a graph

5.8 Behavior of Functions

5.8.1 Understand and compare the behavior of linear, quadratic, and exponential functions

5.8.2 Observe that exponential growth eventually exceeds polynomial growth

5.8.3 Use properties of exponents to interpret expressions for exponential functions

5.9 Constructing Functions

5.9.1 Construct linear and exponential functions given graphs, descriptions, or two input-output pairs

5.10 Composite Functions

5.10.1 Simplify expressions for composite functions such as f(g(x))

5.11 Inverse Functions

5.11.1 Find and interpret inverse functions

5.12 Transformations of Functions

5.12.1 Describe changes to the graph of y = f(x) when transformed as y = f(x) + k, y = kf(x), and y = f(x +

5.13 Graphing Inequalities

5.13.1 Graph the solutions to a linear inequality in two variables as a half-plane

5.13.2 Graph the solution set of a system of inequalities as the intersection of corresponding half-planes

6. Number

6.1 Units and Estimations

6.1.1 Converting between units (e.g., currency, time, length)

6.1.2 Understanding units in problems

6.1.3 Choosing appropriate levels of accuracy

6.2 Time Calculations

6.2.1 Units of time: seconds, minutes, hours, days, months, years

6.2.2 24-hour and 12-hour clock formats

6.3 Speed, Distance, and Time Problems

6.3.1 Understanding and solving problems related to motion

6.3.2 Using the speed formula correctly

6.4 Number Types and Operations

6.4.1 Natural numbers, integers, prime numbers, square numbers

6.4.2 Rational and irrational numbers, real numbers

6.4.3 Sum or product of two rational numbers is rational

6.4.4 Sum of a rational and an irrational number is irrational

6.4.5 Product of a non-zero rational and an irrational number is irrational

6.4.6 Symbols: =, ≠, ⩽, ⩾, <, >

6.5 Basic Arithmetic Operations

6.5.1 Four operations and parentheses

6.5.2 Applies to integers, fractions, and decimals

6.6 Multiples and Factors

6.6.1 Greatest Common Factor (GCF)

6.6.2 Least Common Multiple (LCM)

6.7 Ratio and Proportion

6.7.1 Understanding and solving problems using ratios

6.7.2 Proportions in real-world contexts

6.8 Fractions, Decimals, and Percentages

6.8.1 Convert between decimals, fractions, ratios, and percentages

6.8.2 Ordering different forms by magnitude

6.9 Percentages and Applications

6.9.1 Calculating percentage increases and decreases

6.9.2 Applications like interest and profit

6.9.3 Includes reverse percentages

6.9.4 Includes both simple and compound interest

6.9.5 Includes percentiles

6.10 Exponents and Scientific Notation

6.10.1 Scientific notation (Standard Form)

6.10.2 Basic exponent rules

6.10.3 Positive, negative, zero, and fractional exponents

6.11 Radicals and Roots

6.11.1 Calculation and simplification of square root and cube root expressions

6.11.2 Simplify radical expressions

7. Geometrical Measurement

7.1 Units of Measurement

7.1.1 Understand and convert between metric units (mm, cm, m, km)

7.1.2 Understand and convert between area and volume units (mm², cm², m², ha, km², mm³, cm³, ml, l, m³)

7.2 Perimeter and Area

7.2.1 Calculate perimeter and area of rectangles and triangles

7.2.2 Calculate area of compound shapes derived from rectangles and triangles

7.2.3 Calculate area of trapezoids and parallelograms

7.3 Circles

7.3.1 Calculate circumference and area of circles

7.3.2 Calculate arc length and area of a sector (sector angles in degrees only)

7.4 Surface Area and Volume

7.4.1 Calculate surface area and volume of prisms and pyramids (including cuboids, cylinders, and cones)

7.4.2 Calculate surface area and volume of spheres

7.5 Compound Shapes

7.5.1 Calculate areas and volumes of compound shapes

7.6 Using Geometric Shapes

7.6.1 Describe objects using geometric shapes and their properties

7.7 Cross Sections and Rotations

7.7.1 Identify the shapes of 2D cross sections of 3D objects

7.7.2 Identify 3D objects generated by rotations of 2D shapes

7.8 Density and Modeling

7.8.1 Apply concepts of density based on area and volume in modeling situations

7.9 Geometric Design

7.9.1 Apply geometric methods to solve design problems

8. Algebra

8.1 Inequalities

8.1.1 Writing, showing, and interpreting inequalities on the real number line

8.1.2 Create and solve linear inequalities

8.2 Linear Expressions and Equations

8.2.1 Create expressions and solve linear equations, including fractional expressions

8.2.2 Explain algebraic steps of a solution

8.2.3 Interpret solutions in a given context

8.3 Exponents and Powers

8.3.1 Rules of exponents, including negative and fractional exponents

8.3.2 Basic exponent calculations

8.4 Rearrangement of Formulae

8.4.1 Rearrange and evaluate formulae, including algebraic manipulation to prove identities

8.4.2 Make a variable the subject of an equation

8.5 Linear Equations in Two Variables

8.5.1 Create and solve systems of linear equations algebraically and graphically

8.6 Identifying Algebraic Components

8.6.1 Identify terms, factors, and coefficients

8.7 Expanding and Simplifying

8.7.1 Expansion of parentheses, including the square of a binomial

8.7.2 Simplify expressions

8.8 Algebraic Fractions

8.8.1 Simplification using factorization

8.8.2 Addition or subtraction of fractions with linear denominators

8.8.3 Multiplication or division and simplification of two fractions

8.9 Quadratic Equations

8.9.1 Create and solve quadratic equations by inspection, factorization, quadratic formula, and completing

8.9.2 Write quadratic expressions in the form (x-a)^2 + b and state the minimum value

8.10 Solving Equations

8.10.1 Solve simple rational and radical equations and identify extraneous solutions

8.11 Sequences and Patterns

8.11.1 Continuation of sequences of numbers or patterns

8.11.2 Recognize patterns and generalize to algebraic statements

8.11.3 Determine the nth term of a sequence

8.11.4 Derive and use the formula for the sum of a finite geometric series

8.12 Direct and Inverse Variation

8.12.1 Express direct and inverse variation in algebraic terms

8.12.2 Use variation formulas to find unknown quantities

8.13 Factorization

8.13.1 Factorization using common factors, difference of squares, trinomials, and four-term expressions

9. Probability

9.1 Basic Probability Concepts

9.1.1 Understand probability P(A) as a fraction, decimal, or percentage

9.1.2 Interpret probability values and their significance

9.1.3 Calculate probability using the rule P(A) = 1 – P(A')

9.2 Experimental Probability

9.2.1 Use relative frequency as an estimate of probability

9.3 Expected Outcomes

9.3.1 Calculate the expected number of occurrences in probability experiments

9.4 Combining Events

9.4.1 Apply the addition rule P(A or B) = P(A) + P(B) – P(A and B)

9.4.2 Apply the multiplication rule P(A and B) = P(A) × P(B)

9.5 Independent Events

9.5.1 Understand and determine if two events are independent

9.6 Possibility Diagrams

9.6.1 Use possibility diagrams to list all outcomes

9.7 Tree Diagrams

9.7.1 Construct and interpret tree diagrams for successive selections with or without replacement

10. Coordinate Geometry

10.1 Plotting Points

10.1.1 Plot points and read coordinates in the Cartesian plane

10.2 Distance Between Points

10.2.1 Calculate the distance between two points using the distance formula

10.3 Midpoint of a Line Segment

10.3.1 Find the midpoint of a line segment

10.4 Partitioning a Line Segment

10.4.1 Find the point on a directed line segment that partitions the segment in a given ratio

10.5 Slope of a Line

10.5.1 Calculate the slope (gradient) of a line segment

10.6 Equation of a Straight Line

10.6.1 Interpret and obtain the equation of a straight line in the form y = mx + b

10.6.2 Interpret and obtain the equation of a straight line in the form ax + by = d (where a, b, and d are

10.6.3 Find the equation of a straight line given two points

10.7 Parallel Lines

10.7.1 Understand and find the slope of parallel lines

10.7.2 Find the equation of a line parallel to a given line that passes through a given point

10.8 Perpendicular Lines

10.8.1 Understand the relationship between slopes of perpendicular lines

10.8.2 Find the equation of a line perpendicular to a given line that passes through a given point

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Tangents from an External Point are Equal in Length

Introduction