Home

Maths

Graphs

Coordinates and ratio

Coordinates and ratio

Videos

Summary

Exercises

Select Lesson

Exam Board

Select an option

Statistics

Sets and Venn diagrams

Sampling and bias

Collecting data: types and classes of data

Mean, median, mode and range

Simple charts and graphs

Pie charts

Scatter graphs

Frequency tables: finding averages

Grouped frequency tables

Box plots - Higher

Cumulative frequency - Higher

Histograms and frequency density - Higher

Interpreting data

Comparing data sets

Probability

Basics of probability

Calculating theoretical probabilities

Probability: Expected and relative frequency

The AND / OR rules

Probability tree diagrams

Conditional probability - Higher

Experimental probability: frequency trees

Trigonometry

Pythagoras' theorem

Sin, cos, tan

Trigonometry: Finding angles and sides

Exact trigonometric values

Sine and cosine rules - Higher

3D Pythagoras - Higher

3D Trigonometry - Higher

Vectors

Vectors - Higher

Angles and geometry

Angles: types, notation and measuring

Basic angle rules

Angles in parallel lines

Circle theorems - Higher

Constructing triangles: SSS, SAS, ASA

Construction: angle and perpendicular bisectors

Construction: Loci

Bearings

Maps and scale drawings

Shapes and area

Properties of 2D shapes

Congruence: conditions for congruent triangles

Similar shapes: Scaling

The four transformations

Area and perimeter: Formulae

Area and circumference of circles: Formulae

3D shapes: faces, edges, vertices

Surface area of 3D shapes: Nets, formulae

Volume of 3D shapes: Formulae

Volume of 3D shapes: Comparing, rates of flow

Area and volume scale factors

Projections and elevations of 3D shapes

Ratio proportion and rates of change

Ratio

Direct and inverse proportion

Finding percentages and percentage change

Compound growth and decay

Converting units: metric and imperial

Converting units: area and volume

Time intervals: converting units of time

Speed, density and pressure: Formulae and units

Graphs

Coordinates and midpoints

Straight line graphs

Drawing straight line graphs

Finding the gradient of a straight line

Equation of a straight line: y = mx + c

Coordinates and ratio

Parallel and perpendicular lines

Quadratic graphs

Reciprocal and cubic graphs

Exponential graphs and circles - Higher

Trigonometric graphs - Higher

Solving equations using graphs

Graph transformations - Higher

Real-life graphs

Distance-time graphs

Velocity-time graphs - Higher

Gradients of real-life graphs - Higher

Algebra

Simplifying algebraic expressions

Multiplying and dividing algebraic expressions

Single brackets: Expanding and factorising

Double brackets: Expanding and factorising

Double and triple brackets - Higher

Solving equations

Expressions, equations, formulae, functions and identities

Writing formulae and equations from word problems

Writing formulae and equations from diagrams

Rearranging formulae

Factorising quadratics

The quadratic formula - Higher

Complete the square - Higher

Algebraic fractions - Higher

Sequences

Finding the nth term

Solving inequalities

Inequalities on graphs - Higher

Iteration - Higher

Simultaneous equations: elimination and substitution

Non-linear simultaneous equations - Higher

Algebraic proof - Higher

Composite and inverse functions - Higher

Number

Types of numbers

Order of operations: BODMAS

Multiplying and dividing by powers of 10

Multiplying and dividing whole numbers

Multiplying and dividing decimals

Negative numbers: add, subtract, multiply, divide

Prime numbers and prime factorisation

Multiples, factors and prime factors

LCM and HCF

Fractions

Fractions, decimals and percentages

Writing recurring decimals as fractions

Rounding: Integers, decimal places, significant figures

Estimation

Error intervals

Upper and lower bounds - Higher

Powers and roots: Square and cube numbers

Laws of indices: multiply, divide, brackets

Index laws: negative and fractional indices - Higher

Surds: Simplify, add and subtract - Higher

Rationalising surds - Higher

Standard form calculations

Explainer Video

Tutor: Toby

Summary

​​Coordinates and ratio

In a nutshell

Knowing the coordinates of the end-points of a line segment is enough to describe locations on the line segment. The midpoint is just halfway between the xxx- and yyy-coordinates, but other positions can be found using ratios.​



Midpoints

​To find the point in the middle of two other points, find the middle between the points' xxx-coordinates and the middle between the points' yyy-coordinates. These middle values are the xxx​- and yyy- coordinates of the midpoint.


Finding the middle between two numbers can be done by halving the difference between them and then adding this to the smaller of the two numbers. Or more formally, using the following procedure.

​

procedure

1.
Designate one of your points as point AAA​ and the other as point BBB​. Let point AAA have the coordinates (xA,yA)(x_A,y_A)(xA​,yA​)​ and point BBB have the coordinates (xB,yB)(x_B,y_B)(xB​,yB​).​
2.
Find the middle xxx-value using the formula:
xA+xB−xA2x_A+\frac{x_B-x_A}{2}xA​+2xB​−xA​​​​
3.
Find the middle yyy-value using the similar formula:
yA+yB−yA2y_A+\frac{y_B-y_A}{2}yA​+2yB​−yA​​​​
4.
You now have that the midpoint has coordinates (xA+xB−xA2,yA+yB−yA2)(x_A+\frac{x_B-x_A}{2},y_A+\frac{y_B-y_A}{2})(xA​+2xB​−xA​​,yA​+2yB​−yA​​).​



Using a ratio to locate a point 

The same idea is used here. Instead of halving the difference between the points' coordinates, you apply the given ratio.


procedure

1.
Designate one of your points as point AAA​ and the other as point BBB​. Let point AAA have the coordinates (xA,yA)(x_A,y_A)(xA​,yA​)​ and point BBB have the coordinates (xB,yB)(x_B,y_B)(xB​,yB​).​
2.
Suppose the new point splits the line from A to B into the ratio α:β\alpha:\betaα:β. This means the new point is αα+β\frac{\alpha}{\alpha+\beta}α+βα​ of the line from A.​
3.
Find the new point's xxx-coordinate using the formula:
xA+αα+β(xB−xA)x_A+\frac{\alpha}{\alpha+\beta}(x_B-x_A)xA​+α+βα​(xB​−xA​)​​
4.
Find the new point's yyy-coordinate using the similar formula:
yA+αα+β(yB−yA)y_A+\frac{\alpha}{\alpha+\beta}(y_B-y_A)yA​+α+βα​(yB​−yA​)​​
5.
You now have that the new point has coordinates (xA+αα+β(xB−xA),yA+αα+β(yB−yA))(x_A+\frac{\alpha}{\alpha+\beta}(x_B-x_A),y_A+\frac{\alpha}{\alpha+\beta}(y_B-y_A))(xA​+α+βα​(xB​−xA​),yA​+α+βα​(yB​−yA​)).​


Example 1

Take the line segment with the points A (2,−8)(2,-8)(2,−8) and B (−6,24)(-6,24)(−6,24) at either end. The point C is located on this line segment such that it divides the line in the ratio 1:31:31:3. What are the coordinates of point C?


The ratio says that point C is a quarter of the distance of the line segment from point A. This is found by dividing the first part of the ratio, 111 by the sum of both parts of the ratio, 1+31+31+3:

​11+3=14\frac{1}{1+3}=\frac141+31​=41​


You want to therefore find the coordinates that are a quarter of the xxx- and yyy-differences from point A.


The difference between the xxx coordinates is 

−6−2=−8-6-2=-8−6−2=−8​​


To apply the ratio, multiply this by the quarter, then add it to the 222:

​−8×14=−2-8\times\frac14=-2−8×41​=−2


2+(−2)=02+(-2)=02+(−2)=0


Thus the xxx-coordinate of point C is 000. Now do the same for the yyy-coordinate: the difference is

24−(−8)=3224-(-8)=3224−(−8)=32


Multiplying this by the quarter (from the ratio) then adding to −8-8−8 gives

32×14=832\times\frac14=832×41​=8


−8+8=0-8+8=0−8+8=0


So the yyy-coordinate of point C is also 000. Hence point C has coordinates (0,0)‾\underline{(0,0)}(0,0)​​.​​

Read more

Learn with Basics

Learn the basics with theory units and practise what you learned with exercise sets!
Length:
Coordinates in the first quadrant

Unit 1

Coordinates in the first quadrant

X and Y coordinates

Unit 2

X and Y coordinates

Jump Ahead

Score 80% to jump directly to the final unit.

Optional

Coordinates and ratio

Unit 3

Coordinates and ratio

Final Test

Test reviewing all units to claim a reward planet.

Create an account to complete the exercises

FAQs - Frequently Asked Questions

How does splitting a line segment into a ratio differ from finding the midpoint?

Instead of halving the difference between the points' coordinates, you apply the given ratio. Technically, finding the midpoint is just splitting the line into the ratio 1:1.

How do you use a ratio to find the scale factor?

If a line AB is split into the ratio α:β, then to go from A to the new point, multiply the difference between A and B by α/(α+β). ​

How do you find the midpoint of two points?

​To find the point in the middle of two other points, find the middle between the points' x-coordinates and the middle between the points' y-coordinates.

©2020 - 2024 evulpo AG

Beta