Partitioning a directed line segment in a specific ratio is a fundamental concept in coordinate geometry, integral to solving various mathematical problems. This topic holds particular significance for students following the Cambridge IGCSE curriculum in Mathematics - US - 0444 - Advanced. Mastery of this concept enhances spatial reasoning and provides a foundation for more complex geometric and algebraic applications.

Key Concepts

Understanding Directed Line Segments

A directed line segment has a distinct direction from one endpoint to another, often referred to as the initial point and the terminal point. This directionality is crucial when discussing ratios, as it determines how the segment is divided. Unlike undirected segments, directed segments consider the orientation, enabling precise calculations in coordinate geometry.

Ratios and Their Significance

A ratio expresses the relationship between two quantities, indicating how many times one value contains another. In the context of line segments, a ratio $ m:n $ signifies that the segment is divided into two parts where the lengths of these parts are in proportion $ m $ to $ n $. This proportionality is essential for determining the exact point of division on the segment.

The Section Formula

The section formula is a pivotal tool in coordinate geometry used to find the coordinates of a point that divides a line segment internally or externally in a given ratio. Consider two points $ A(x_1, y_1) $ and $ B(x_2, y_2) $. The point $ P(x, y) $ that divides the segment $ AB $ internally in the ratio $ m:n $ is given by:

$$ x = \frac{mx_2 + nx_1}{m + n}, \quad y = \frac{my_2 + ny_1}{m + n} $$

For external division, the formulas adjust to:

$$ x = \frac{mx_2 - nx_1}{m - n}, \quad y = \frac{my_2 - ny_1}{m - n} $$

These formulas are instrumental in locating the precise point of division based on the specified ratio.

Internal vs. External Division

Internal division occurs when the point of division lies between the two endpoints of the segment, adhering to the given ratio $ m:n $. External division, on the other hand, places the dividing point outside the segment, effectively extending the line beyond one of the endpoints. Understanding the distinction between these two types of division is essential for accurate application of the section formula.

Applications of Partitioning in Coordinate Geometry

Partitioning a line segment is not just a theoretical exercise; it has practical applications in various fields such as engineering, architecture, computer graphics, and robotics. For instance, in computer graphics, dividing line segments accurately is crucial for rendering shapes and animations. In engineering, precise measurements and divisions are necessary for designing structural components.

Examples and Problem-Solving

**Example 1:** Find the point that divides the line segment joining $ A(2, 3) $ and $ B(8, 7) $ internally in the ratio $ 2:3 $.

**Solution:** Applying the section formula for internal division:

$$ x = \frac{2 \times 8 + 3 \times 2}{2 + 3} = \frac{16 + 6}{5} = \frac{22}{5} = 4.4 $$ $$ y = \frac{2 \times 7 + 3 \times 3}{2 + 3} = \frac{14 + 9}{5} = \frac{23}{5} = 4.6 $$

Therefore, the coordinates of the dividing point $ P $ are $ (4.4, 4.6) $.

Graphical Interpretation

Graphing the scenario helps in visualizing the division of the segment. Plotting points $ A $ and $ B $ on a coordinate plane and drawing the segment $ AB $, the point $ P $ can be located using the calculated coordinates. This visual aid reinforces the conceptual understanding of partitioning a directed line segment.

Common Mistakes to Avoid

Misapplying the section formula by confusing internal and external division.
Incorrectly substituting values into the formula, leading to calculation errors.
Ignoring the directionality of the line segment, which can affect the outcome in directed contexts.

Practice Problems

Find the point that divides the segment joining $ C(1, 2) $ and $ D(5, 10) $ externally in the ratio $ 3:2 $.
A point $ Q $ divides the line segment from $ E(-3, 4) $ to $ F(9, -2) $ in the ratio $ 4:5 $. Determine the coordinates of $ Q $.
Given points $ G(0, 0) $ and $ H(6, 8) $, find the point that partitions $ GH $ internally in the ratio $ 1:1 $.

Attempting these problems will solidify your understanding of the section formula and its applications in partitioning directed line segments.

Advanced Concepts

Theoretical Foundations of the Section Formula

The section formula is underpinned by vector algebra and the principles of weighted averages. By treating coordinates as vectors, the formula effectively computes a weighted mean of the positions of the two endpoints. This approach ensures that the dividing point lies proportionally closer to one endpoint based on the specified ratio, maintaining consistency across various applications.

Mathematical Derivation of the Section Formula

Consider points $ A(x_1, y_1) $ and $ B(x_2, y_2) $, and let $ P(x, y) $ divide $ AB $ in the ratio $ m:n $. The derivation begins by expressing the position vectors of $ A $, $ B $, and $ P $. By setting up the equation based on the ratio, solving for $ x $ and $ y $ leads to the section formula:

$$ x = \frac{mx_2 + nx_1}{m + n}, \quad y = \frac{my_2 + ny_1}{m + n} $$

This derivation showcases the fundamental relationship between the coordinates of the endpoints and the dividing point.

Applications in Higher-Dimensional Geometry

The principles of partitioning a line segment extend beyond two dimensions. In higher-dimensional spaces, the section formula can be generalized to include additional coordinates, allowing for the determination of dividing points in three-dimensional geometry and beyond. This generalization is vital in fields such as computer-aided design (CAD) and physics, where multidimensional modeling is commonplace.

Integration with Vector Spaces

In vector spaces, the section formula can be expressed using vector notation, enhancing its applicability in various mathematical disciplines. Representing points as vectors facilitates operations such as scaling and addition, which are integral to more advanced topics like linear transformations and vector calculus.

Complex Problem-Solving Scenarios

**Problem:** Given three points $ A(2, 3) $, $ B(4, 6) $, and $ C(6, 9) $, determine if point $ P(5, 7.5) $ lies on the line segment $ AC $ and find the ratio in which it divides the segment.

**Solution:** First, verify if $ P $ lies on $ AC $. Calculate the slope of $ AC $ and the slope of $ AP $. If they are equal, $ P $ lies on $ AC $.

$$ \text{Slope of } AC = \frac{9 - 3}{6 - 2} = \frac{6}{4} = 1.5 $$ $$ \text{Slope of } AP = \frac{7.5 - 3}{5 - 2} = \frac{4.5}{3} = 1.5 $$

Since the slopes are equal, $ P $ lies on $ AC $. To find the ratio:

$$ \frac{AP}{PC} = \frac{\sqrt{(5-2)^2 + (7.5-3)^2}}{\sqrt{(6-5)^2 + (9-7.5)^2}} = \frac{\sqrt{9 + 20.25}}{\sqrt{1 + 2.25}} = \frac{\sqrt{29.25}}{\sqrt{3.25}} \approx \frac{5.408}{1.803} \approx 3:1 $$>

Thus, point $ P $ divides $ AC $ in the ratio $ 3:1 $.

Interdisciplinary Connections

Partitioning line segments is not limited to pure mathematics. In physics, it is used in vector analysis and dynamics to resolve forces acting at different points. In computer graphics, accurately dividing segments is essential for rendering objects and animations. Additionally, in economics, similar principles apply in market analysis and optimization problems where proportional divisions represent resource allocations.

Exploring Non-Euclidean Geometries

While the section formula is primarily used in Euclidean geometry, exploring its applications in non-Euclidean contexts, such as spherical or hyperbolic geometry, presents advanced challenges. Understanding how partitioning behaves under different geometric frameworks broadens the scope of its applicability and deepens mathematical insight.

Advanced Theorems and Proofs

Delving into advanced theorems related to partitioning, such as Menelaus's Theorem and Ceva's Theorem, provides a deeper theoretical foundation. These theorems extend the concept of dividing line segments to more complex geometric configurations, enabling the solving of intricate problems involving concurrent lines and transversals.

Utilizing Technology for Enhanced Learning

Modern educational tools, including dynamic geometry software and graphing calculators, facilitate a better understanding of partitioning concepts. Visualizing the division of segments and manipulating ratios in real-time enhances comprehension and allows for experimentation beyond traditional methods.

Comparison Table

Aspect	Internal Division	External Division
Definition	Point lies between the two endpoints of the segment.	Point lies outside the segment, extending beyond one endpoint.
Ratio Representation	Positive ratio $ m:n $.	Positive ratio $ m:n $, but applied as $ m:-n $ in formulas.
Section Formula	$$x = \frac{mx_2 + nx_1}{m + n}$$ $$y = \frac{my_2 + ny_1}{m + n}$$	$$x = \frac{mx_2 - nx_1}{m - n}$$ $$y = \frac{my_2 - ny_1}{m - n}$$
Graphical Representation	Point lies on the line segment between endpoints.	Point extends the line segment beyond one endpoint.
Applications	Determining midpoints, dividing segments for construction.	Extending lines, solving for points in advanced geometric problems.

Summary and Key Takeaways

Partitioning a directed line segment involves dividing it in a specified ratio using the section formula.
The section formula varies for internal and external division, crucial for accurate calculations.
Understanding the theoretical foundations enhances problem-solving and interdisciplinary applications.
Visualization and technological tools aid in comprehending and applying partitioning concepts effectively.
Mastery of this topic is essential for advanced studies in coordinate geometry and related mathematical fields.

Examiner Tip

Tips

• **Memorize the Section Formula:** Keep both internal and external formulas at your fingertips to apply them swiftly during exams.

• **Visualize the Problem:** Drawing a diagram can help in understanding the division type and applying the correct formula.

• **Check Your Ratios:** Always verify whether the division is internal or external to choose the appropriate formula.

• **Use Mnemonics:** Remember "Internal Includes" (Internal division formulas include positive ratios) and "External Extends" (External division formulas involve subtraction).

• **Practice Regularly:** Consistent problem-solving will reinforce your understanding and speed in applying the section formula.

Did You Know

1. The concept of partitioning line segments dates back to ancient Greek mathematics, where Euclid extensively used it in his geometric proofs.

2. In computer graphics, algorithms for dividing segments in precise ratios are fundamental for creating smooth animations and realistic models.

3. Partitioning principles are applied in robotics for path planning, allowing robots to navigate through precise coordinates efficiently.

Common Mistakes

1. **Confusing Internal and External Division:** Students often mix up the formulas for internal and external division, leading to incorrect coordinates.
Incorrect: Using external division formula for an internal division problem.
Correct: Applying the internal division formula when the point lies between the endpoints.

2. **Sign Errors in Ratios:** Forgetting to adjust the ratio signs for external division can result in wrong calculations.
Incorrect: Using positive ratios for external division without considering subtraction.
Correct: Incorporating negative ratios in the formula for external division.

3. **Ignoring Directionality:** Overlooking the direction of the line segment can affect the outcome, especially in directed contexts.
Incorrect: Treating the segment as undirected when direction matters.
Correct: Considering the initial and terminal points to maintain accurate orientation.

FAQ

What is the section formula?

The section formula calculates the coordinates of a point that divides a line segment internally or externally in a given ratio using the coordinates of the segment's endpoints.

How do you determine if a point divides a segment internally or externally?

If the dividing point lies between the two endpoints, it is internal division. If it lies outside the segment, extending the line beyond an endpoint, it is external division.

Can the section formula be used in three dimensions?

Yes, the section formula can be extended to three dimensions by including the z-coordinates, allowing you to find the dividing point in 3D space.

What are some real-world applications of partitioning line segments?

Partitioning line segments is used in engineering for designing components, in computer graphics for rendering images, and in robotics for path planning, among other applications.

How do vectors relate to the section formula?

Vectors provide a basis for understanding the section formula as they represent points in space, allowing the formula to be interpreted as a weighted average of the vectors of the endpoints.

What is the difference between a ratio and a proportion in partitioning?

A ratio describes the relative sizes of two segments, while a proportion states that two ratios are equal. In partitioning, we use ratios to define how a segment is divided, and proportions can be used to compare different partitioning scenarios.

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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Find the Point on a Directed Line Segment that Partitions the Segment in a Given Ratio

Introduction