DP Maths AI HL 3.1 Coordinate Geometry

DP Maths AI HL 3.1 Coordinate Geometry - Full notes

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DP Maths AI HL | 3.1 Coordinate Geometry

DP MATHEMATICS AI · TOPIC 3: GEOMETRY AND TRIGONOMETRY

3.1

Coordinate Geometry

Full notes for Applications and Interpretation HL

Contents

1	Coordinates in two and three dimensions
2	Distance between two points
3	Distance in three dimensions
4	Midpoints
5	Gradients and equations of lines
6	Perpendicular bisectors
7	Voronoi diagrams: construction
8	Voronoi diagrams: interpretation
9	Adding a site and the largest empty circle
10	Volumes of three-dimensional solids
11	Surface areas of three-dimensional solids
12	Composite solids
13	Angles in three dimensions
14	Angle between a line and a plane
15	Approximation, accuracy and percentage error
16	Applications: maps, scales and design
17	Technology for geometry
18	Mixed review: worked problems
19	Vectors in two and three dimensions (HL)
20	Vector equations of lines (HL)
21	Investigation: placing a new fire station
22	Exam-style questions with markscheme
23	Common misconceptions and links
24	Check your understanding
25	Summary and topic map
26	Answers to practice exercises

These are full teaching notes. Read each section in order, try the Think about it question before you read on, and work through every worked example before attempting the practice exercise.

Before you start

Rate yourself on each statement now, and again after you've worked through the notes. Shade R (not yet), A (getting there) or G (confident).

I can…	R	A	G
find distances and midpoints in two and three dimensions
find gradients, equations of lines and perpendicular bisectors
construct and interpret Voronoi diagrams, including adding a site
solve the largest empty circle (toxic waste dump) problem
calculate volumes and surface areas of prisms, pyramids, cylinders, cones and spheres
find angles between lines and planes in 3D
use approximation, significant figures and percentage error

HL booklet: this booklet contains all the SL content plus an introduction to vectors and vector equations of lines.

Guiding questions

How can coordinates describe positions and distances in the real world? How can geometry help decide where to put a new facility?

How this topic is examined

Coordinate geometry questions in AI include Voronoi diagrams (constructing edges with perpendicular bisectors, interpreting regions, adding a new site, finding the point furthest from all sites), 3D distances and angles, and volume and surface area in context.

Show the equations of perpendicular bisectors, and give lengths and angles to 3 s.f. with units. Use percentage error to judge the accuracy of estimates.

Mathematics: Applications and Interpretation places strong emphasis on technology, modelling and interpretation. Most papers allow a GDC, so you are expected to use it efficiently, record the method you used, and interpret every answer in context.

Command term	What the examiner wants
Write down	an answer with little or no working; it should be obtainable by inspection
Find / Calculate / Determine	obtain the answer showing relevant working, including what you entered into a GDC
Show that	obtain the given result; every step must be shown and the result must not be assumed
Estimate	obtain an approximate value
Sketch	a diagram showing the general shape and key features, labelled
Interpret / Comment	explain the meaning of a result in context, or give a judgement
Justify / Explain	give valid reasons or evidence
Hence	use the previous result

How to use these notes

Box	What it's for
Think about it	a question to get you thinking before a section
Why it works	a derivation or explanation of a key result
Key point	a result or idea you must not forget
Common mistake	an error examiners see every year, and how to avoid it
GDC tip	how to use your graphic display calculator efficiently
In the real world	where the mathematics is used outside the classroom
Worked example	a full solution with IB-style marks (M1 method, A1 answer, R1 reasoning, AG answer given), graded basic, standard and exam level
Practice exercise	graded questions for each section; full answers are at the end of the booklet

1 Coordinates in two and three dimensions

Locating points.

Think about it

A drone hovers 30 m above a point that is 40 m east and 20 m north of its pilot. How far is the drone from the pilot?

In two dimensions a point has coordinates (x, y); in three dimensions (x, y, z), with the z-axis perpendicular to the xy-plane. Coordinates turn geometric questions into algebra: distances, midpoints, gradients and angles can all be calculated.

2 Distance between two points

Pythagoras in coordinates.

WORKED EXAMPLE 1 2D distance (basic)

Find the distance between (−3, 4) and (5, −2).

Solution

√(82 + 62)M1

= 10A1

Practice exercise 1

1. Find the distance between (1, 1) and (7, 9).

2. Find k > 0 if the distance between (2, k) and (5, 1) is 5.

3 Distance in three dimensions

Pythagoras twice.

WORKED EXAMPLE 2 Drone distance (standard)

A drone is at (40, 20, 30) relative to its pilot at the origin (metres). Find its distance.

Solution

√(1600 + 400 + 900) = √2900M1

= 53.9 mA1

Practice exercise 2

1. Find the distance between (1, −2, 3) and (4, 2, 15).

2. Find the length of the space diagonal of a 3 × 4 × 12 box.

4 Midpoints

Averages of coordinates.

Practice exercise 3

1. Find the midpoint of (−4, 6) and (8, 2).

2. M(3, 1, 5) is the midpoint of A(1, −2, 4) and B. Find B.

5 Gradients and equations of lines

Review.

The gradient m = (y2 − y1)/(x2 − x1). Lines: y = mx + c, y − y1 = m(x − x1), ax + by + d = 0. Perpendicular gradients multiply to −1.

Practice exercise 4

1. Find the line through (2, 3) and (6, 11).

2. Find the gradient perpendicular to the line 2x − 5y = 1.

6 Perpendicular bisectors

Points equidistant from two sites.

The perpendicular bisector of AB is the set of points equidistant from A and B. It passes through the midpoint of AB, perpendicular to AB. To find its equation: find the midpoint, find the gradient of AB, take the negative reciprocal, and write the line through the midpoint.

WORKED EXAMPLE 3 Perpendicular bisector (standard)

Find the perpendicular bisector of A(2, 1) and B(6, 7).

Solution

Midpoint (4, 4); gradient AB = 3/2A1

Perpendicular gradient −2/3A1

y − 4 = −(2/3)(x − 4): 2x + 3y − 20 = 0M1 A1

Practice exercise 5

1. Find the perpendicular bisector of (0, 0) and (6, 4).

2. Show that (5, 1) is equidistant from (2, 3) and (8, 3).

7 Voronoi diagrams: construction

Regions of nearest sites.

Given a set of points called sites (for example, hospitals or mobile phone masts), a Voronoi diagram divides the plane into cells: each cell contains all points that are closer to its site than to any other. The boundaries (edges) are parts of perpendicular bisectors between neighbouring sites, and the points where three or more edges meet are vertices, equidistant from three or more sites.

To construct by hand: draw the perpendicular bisectors between neighbouring sites, and keep only the parts that bound each site's cell. In examinations, diagrams are usually partly drawn, and you complete an edge or find its equation.

Figure 1. A Voronoi diagram for four sites A(2, 2), B(7, 3), C(4, 7) and D(8, 8).

WORKED EXAMPLE 4 Edge between two sites (standard)

Find the equation of the edge between cells A(2, 2) and B(7, 3).

Solution

Midpoint (4.5, 2.5); gradient AB = 1/5A1

Perpendicular gradient −5A1

y = −5x + 25A1

Practice exercise 6

1. Find the equation of the edge between C(4, 7) and D(8, 8).

2. Explain why a vertex of a Voronoi diagram is equidistant from three sites.

8 Voronoi diagrams: interpretation

Nearest neighbour and service areas.

A point lies in the cell of its nearest site. Voronoi diagrams answer questions such as: which hospital is nearest to a village; how large is each shop's catchment area; which weather station's readings should represent a location. The area of a cell can be found by splitting it into triangles and trapezia.

WORKED EXAMPLE 5 Nearest site (standard)

Using the diagram, which site is nearest to P(5, 5)?

Solution

Distances: PA = √18 = 4.24, PB = √8 = 2.83, PC = √5 = 2.24, PD = √18 = 4.24M1

C is nearestA1

Practice exercise 7

1. Which site is nearest to (8, 1)?

2. Explain how a Voronoi diagram could help a council plan school catchment areas.

9 Adding a site and the largest empty circle

Updating diagrams and the toxic waste dump problem.

Adding a site: draw the perpendicular bisectors between the new site and its neighbours, and erase the parts of old edges now inside the new cell. Largest empty circle (toxic waste dump problem): the point within a boundary that is furthest from every site lies at a vertex of the Voronoi diagram (or on the boundary). Its distance to the nearest site is the radius of the largest circle containing no site.

WORKED EXAMPLE 6 Toxic waste dump (exam level)

A vertex V of the diagram is equidistant from A(2, 2), B(7, 3) and C(4, 7). Find V and the distance to the nearest site.

Solution

Edge AB: y = −5x + 25; edge AC: midpoint (3, 4.5), gradient of AC = 5/2, so y = −0.4x + 5.7M1 A1

Solve −5x + 25 = −0.4x + 5.7: x = 4.20, y = 4.02M1

V(4.20, 4.02), distance to A = √(2.202 + 2.022) = 2.98 kmA1 A1

Practice exercise 8

1. Find the circumcentre of triangle (0, 0), (6, 0), (0, 8) and the radius of the largest empty circle centred there.

2. Why might the best location for an unpleasant facility be at a Voronoi vertex?

10 Volumes of three-dimensional solids

Formulas and context.

Solid	Volume	Surface area
Cuboid	lwh	2(lw + lh + wh)
Prism	cross-section area × length	sum of faces
Cylinder	πr2h	2πr2 + 2πrh
Pyramid	⅓ × base area × h	base + triangles
Cone	⅓πr2h	πr2 + πrl
Sphere	(4/3)πr3	4πr2
Hemisphere	(2/3)πr3	3πr2

WORKED EXAMPLE 7 Volume in context (standard)

A cylindrical tank has diameter 2.4 m and height 3.5 m. Find its capacity in litres.

Solution

V = π(1.2)2(3.5) = 15.8 m3M1 A1

= 15 800 litresA1

Practice exercise 9

1. Find the volume of a cone with radius 5 cm and height 12 cm.

2. A sphere has volume 500 cm3. Find its radius.

11 Surface areas of three-dimensional solids

Area of the outside.

Surface area counts every face. For a cone, the curved area is πrl, where the slant height l = √(r2 + h2). For an open container, omit the missing face.

WORKED EXAMPLE 8 Cone surface area (standard)

Find the total surface area of a cone with r = 6 cm and h = 8 cm.

Solution

l = 10A1

π(36) + π(6)(10) = 96π = 302 cm2A1

Practice exercise 10

1. Find the surface area of a closed cylinder with r = 3 cm, h = 10 cm.

2. Find the surface area of a hemisphere (including the flat face) of radius 4 cm.

12 Composite solids

Add or subtract parts.

WORKED EXAMPLE 9 Silo (exam level)

A grain silo is a cylinder (r = 3 m, h = 10 m) topped by a hemisphere. Find its volume and the area to be painted (excluding the base).

Solution

V = π(9)(10) + (2/3)π(27) = 108π = 339 m3M1 A1

Painted area = 2π(3)(10) + 2π(9) = 78π = 245 m2M1 A1

Practice exercise 11

1. A cube of side 10 cm has a cylindrical hole of radius 2 cm drilled through it. Find the remaining volume.

13 Angles in three dimensions

Find a right-angled triangle.

To find an angle in a 3D shape, identify a right-angled triangle containing it, find the lengths of two sides (often with Pythagoras in 3D), and use trigonometry.

WORKED EXAMPLE 10 Angle in a cuboid (standard)

A cuboid is 6 × 4 × 3. Find the angle between the space diagonal and the base.

Solution

Base diagonal = √52 = 7.21A1

tan θ = 3/7.21M1

θ = 22.6°A1

14 Angle between a line and a plane

Use the projection.

The angle between a line and a plane is the angle between the line and its projection (shadow) onto the plane. Drop a perpendicular from the top of the line to the plane to form the right-angled triangle.

WORKED EXAMPLE 11 Pyramid (exam level)

A square-based pyramid has base side 8 m and vertical height 10 m. Find the angle between a slant edge and the base.

Solution

Half the base diagonal = 4√2 = 5.66A1

tan θ = 10/5.66M1

θ = 60.5°A1

Practice exercise 12

1. Find the angle between a slant face and the base of the pyramid above.

2. A 5 m ladder against a wall reaches 4.5 m high. Find the angle with the ground.

15 Approximation, accuracy and percentage error

How good is an estimate?

vA is the approximate value and vE the exact value. Round only final answers; use 3 significant figures unless told otherwise. Upper and lower bounds of a measurement are half a unit either side of the recorded value.

WORKED EXAMPLE 12 Percentage error (standard)

A student estimates the volume of a ball of radius 11.2 cm as 6000 cm3. Find the percentage error.

Solution

Exact V = (4/3)π(11.2)3 = 5885A1

ε = |6000 − 5885|/5885 × 100 = 1.95%M1 A1

Practice exercise 13

1. Round 0.004 876 to 2 s.f. and 34 562 to 3 s.f.

2. A length is 12.4 cm to 1 d.p. State its bounds.

16 Applications: maps, scales and design

Real-world coordinates.

Maps use coordinate grids and scales: a scale of 1 : 25 000 means 1 cm represents 250 m. Architects and engineers use 3D coordinates to compute lengths of beams, cables and pipes. GPS coordinates locate positions on Earth.

Practice exercise 14

1. Two towns are 7.4 cm apart on a 1 : 50 000 map. Find the real distance.

2. A cable runs from (0, 0, 12) to (9, 12, 0) metres. Find its length.

17 Technology for geometry

Dynamic geometry software.

GDC tip

Use GeoGebra or similar to plot sites, construct perpendicular bisectors and Voronoi diagrams, and measure areas. On a GDC, solve simultaneous equations to find vertices, and use the distance formula in a list for many points.

18 Mixed review: worked problems

Bringing the ideas together.

WORKED EXAMPLE 13 Mixed problem 1 (exam level)

Sites P(0, 0), Q(8, 0), R(4, 6). Find the vertex of the Voronoi diagram and the radius of the largest empty circle centred there.

Solution

Bisector of PQ: x = 4A1

Bisector of PR: midpoint (2, 3), gradient −2/3: y = −(2/3)x + 13/3M1

x = 4: y = 5/3; radius = √(16 + 25/9) = 4.33A1 A1

Practice exercise 15

1. A cone and a cylinder have the same base radius and height. What fraction of the cylinder's volume is the cone's?

19 Vectors in two and three dimensions (HL)

Magnitude and direction.

A vector has magnitude and direction, written as a column or as ai + bj + ck. The magnitude |v| = √(a2 + b2 + c2). The displacement from A to B is AB = b − a. Vectors add head to tail and scale by multiplying components.

WORKED EXAMPLE 14 Displacement (standard)

A plane flies from A(2, 1, 0) to B(8, 9, 3) (km). Find AB and its length.

Solution

AB = (6, 8, 3)A1

|AB| = √109 = 10.4 kmA1

Practice exercise 16

1. Find |3i − 4j + 12k|.

2. Find a unit vector in the direction of (2, −1, 2).

20 Vector equations of lines (HL)

r = a + λb.

A line through the point with position vector a in direction b has equation r = a + λb. In kinematics, r = a + tv gives the position at time t of an object starting at a with constant velocity v.

WORKED EXAMPLE 15 Ship position (standard)

A ship starts at (3, −2) km and moves with velocity (4, 5) km/h. Find its position after 2 hours and its speed.

Solution

r = (3, −2) + 2(4, 5) = (11, 8)A1

Speed √41 = 6.40 km/hA1

Practice exercise 17

1. Find the line through (1, 2, 3) and (4, 0, 7) in vector form.

21 Investigation: placing a new fire station

A Paper 3 style exploration.

A town has fire stations at A(1, 2), B(6, 1) and C(4, 7) (km). A new station is to be added. 1. Draw the Voronoi diagram for A, B and C. 2. Find the vertex equidistant from all three stations and its distance from them. 3. Explain why a new station at this vertex would reduce the maximum response distance in the region near it. 4. Add the new site D at the vertex and describe how the cells change. 5. Discuss other factors (roads, population density) that the model ignores.

Answers. 2. Solving two perpendicular bisectors gives about (3.93, 3.64), at a distance of about 3.36 km from each station. 3. It is the point in the triangle furthest from all existing stations. 4. D takes a cell around the old vertex; the cells of A, B and C shrink. 5. Travel time depends on roads, not straight-line distance; demand varies with population.

22 Exam-style questions with markscheme

Attempt each question under timed conditions before reading the markscheme.

Question 1 (Paper 1 style) (about 10 minutes)

Points A(1, 3) and B(7, 5) are two sites in a Voronoi diagram.

(a) Find the midpoint of AB. [1]

(b) Find the equation of the perpendicular bisector of AB. [3]

Markscheme

Part	Answer	Marks
(a)	(4, 4)	A1
(b)	gradient AB = 1/3	A1
	gradient −3	A1
	y = −3x + 16	A1
(c)	PA = √17, PB = √13	M1
	B	A1

Examiner's comments. Alternatively, test which side of the bisector P lies on.

Question 2 (Paper 2 style) (about 12 minutes)

A water tank is a cylinder of radius 1.5 m and height 4 m with a cone of the same radius and height 1 m on top.

(a) Find the volume of the tank. [3]

(b) Find the external surface area excluding the base. [3]

Markscheme

Part	Answer	Marks
(a)	π(2.25)(4) + (1/3)π(2.25)(1)	M1
	9π + 0.75π = 9.75π	A1
	30.6 m3	A1
(b)	l = √(2.25 + 1) = 1.80	A1
	2π(1.5)(4) + π(1.5)(1.80)	M1
	46.2 m2	A1
(c)	\|30 − 30.63\|/30.63 × 100	M1
	2.06%	A1

Examiner's comments. Use the unrounded volume in (c).

Question 3 (Paper 2 style) (about 10 minutes)

A tent is a square-based pyramid with base 3 m and vertical height 2.2 m.

(a) Find the length of a slant edge. [2]

(b) Find the angle between a slant edge and the base. [2]

Markscheme

Part	Answer	Marks
(a)	√(2.22 + 2.122)	M1
	3.06 m	A1
(b)	tan θ = 2.2/2.12	M1
	46.0°	A1
(c)	(1/3)(9)(2.2)	M1
	6.6 m3	A1

Examiner's comments. Half the base diagonal is 1.5√2 = 2.12 m.

23 Common misconceptions and links

Students often think…	Actually…
the perpendicular bisector passes through A	it passes through the midpoint
the largest empty circle is centred at a site	at a Voronoi vertex or boundary
angle with a plane uses any line	use the projection
percentage error uses the estimate as denominator	use the exact value

How 3.1 connects to the rest of the course

Topic	Connection
3.2 Trigonometry	angles in triangles
2.2 Functions	lines
3.10 Vectors (HL)	displacement
1.1 Approximation	significant figures

24 Check your understanding

Section A: multiple choice

1. The distance from (0, 0, 0) to (2, 3, 6) is

A. 7 B. 11 C. √11 D. 49

2. A Voronoi edge is part of

A. an angle bisector B. a median C. a perpendicular bisector D. a tangent

3. The volume of a sphere is

A. 4πr2 B. (4/3)πr3 C. πr2h D. (1/3)πr3

4. Percentage error of 9.8 for exact 10 is

A. 0.2% B. 20% C. 98% D. 2%

5. A Voronoi vertex is equidistant from

A. two sites B. three or more sites C. the origin D. the edges

Section B: structured

6. Find the perpendicular bisector of (−2, 4) and (4, 0). [3]

7. Find the surface area of a sphere of volume 1000 cm3. [3]

Answers

1. A 2. C 3. B 4. D 5. B

6. midpoint (1, 2), gradient 3/2: y = 1.5x + 0.5

7. r = 6.20: 483 cm2

25 Summary and topic map

Idea	Remember
Distance	√(Δx2 + Δy2 (+ Δz2))
Midpoint	average coordinates
Perpendicular bisector	through midpoint, gradient −1/m
Voronoi	cells of nearest site; edges on bisectors; vertices equidistant
Solids	volume and surface area formulas
3D angles	right-angled triangles; projections
Error	\|vA − vE\|/\|vE\| × 100%

Before the exam, make sure you…

•show bisector equations
•use vertices for largest empty circle
•keep units
•use exact values before rounding

Coordinate Geometry on one page

Cover this page with a sheet of paper and try to rebuild it from memory. Whatever you can't recall is what to revise next.

Figure 2. Topic map for Coordinate Geometry.

26 Answers to practice exercises

Check your working against these answers. Full methods are expected in the exam.

Practice exercise 1

1. 10

2. k = 5

Practice exercise 2

1. 13

2. 13

Practice exercise 3

1. (2, 4)

2. (5, 4, 6)

Practice exercise 4

1. y = 2x − 1

2. −5/2

Practice exercise 5

1. 3x + 2y − 13 = 0

2. both distances √13

Practice exercise 6

1. midpoint (6, 7.5), gradient −4: y = −4x + 31.5

2. it lies on three perpendicular bisectors

Practice exercise 7

1. B

2. each household is assigned to its nearest school

Practice exercise 8

1. (3, 4); radius 5

2. it is as far as possible from the nearest sites

Practice exercise 9

1. 314 cm3

2. 4.92 cm

Practice exercise 10

1. 245 cm2

2. 151 cm2

Practice exercise 11

1. 1000 − 40π = 874 cm3

Practice exercise 12

1. tan θ = 10/4: 68.2°

2. 64.2°

Practice exercise 13

1. 0.0049; 34 600

2. 12.35 ≤ L < 12.45

Practice exercise 14

1. 3.7 km

2. √369 = 19.2 m

Practice exercise 15

1. 1/3

Practice exercise 16

1. 13

2. (2, −1, 2)/3

Practice exercise 17

1. r = (1, 2, 3) + λ(3, −2, 4)

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