coordinate geometry – outcomeslawlessteaching.eu/maths/lconotes/coordinate... · solve problems...

14/09/2018

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Coordinate Geometry – Outcomes Plot and read points on the coordinate plane.

Solve problems about distances between points.

Solve problems about midpoints of line segments.

Solve problems about slopes of lines.

Solve problems about parallel and perpendicular slopes.

Solve problems about distance between points and lengths of line segments.

Solve problems about equations of lines.

Find the point of intersection of two lines graphically.

Solve problems about areas of triangles.

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Plot and Read Points Recall the coordinate

plane:

The x-axis is a horizontal number line with positive numbers to the right and negative numbers to the left.

The y-axis is a vertical number line with positive numbers towards the top and

negative numbers towards the bottom.

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Plot and Read Points

Points on the coordinate plane have two parts – an x-coordinate and a y-coordinate.

The x-coordinate is how far left / right the point is.

The y-coordinate is how far up / down the point is.

Points are always written (x, y).

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Plot and Read PointsWrite down the

coordinates of each point shown in the

diagram:

A = (2, 3)

B = (-2, 3)

C = (-3, 1)

D = (3, -1)

E = (4, 2)

F = (-1, -3)

G = (-1, -3)

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Plot and Read Points Draw a coordinate plane. Plot and label each of the

following points on it:

A = (2, 5)

B = (2, 0)

C = (3, 1)

D = (-3, 1)

E = (-4, -2)

F = (4, -2)

G = (0, 2)

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Solve Problems about Midpoints

A midpoint of a line segment is exactly halfway between its ends.

Its coordinates are the average of the end coordinates.

e.g. Find the midpoint of (3, 5) and (1, 7).

Midpoint = 3+1

2,5+7

2

=4

2,12

2

= (2, 6)

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Solve Problems about Midpoints In the Formulae & Tables book, midpoint is given as:

For points 𝑃 𝑥1, 𝑦1 and 𝑄 𝑥2, 𝑦2 ,

𝑀𝑖𝑑𝑝𝑜𝑖𝑛𝑡 =𝑥1+𝑥2

2,𝑦1+𝑦2

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Find the midpoint of the following pairs of points:

1. A(5, 0), B(1, 4)

2. C(4, 2), D(7, 6)

3. E(-9, 3), F(7, -7)

4. G(8, 1), H(-2, -5)

5. I(4, -1), J(-5, 9)

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Solve Problems about MidpointsGiven one endpoint and the midpoint, find the other

endpoint:

1. Endpoint (0, 0), midpoint (2, 4)

2. Endpoint (1, 3), midpoint (3, 6)

3. Endpoint (-6, 9), midpoint (2, -4)

4. Endpoint (6, -4), midpoint (5, -1)

5. Endpoint (3, -8), midpoint (1, -6)

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Solve Problems about Midpoints

Plot the points 𝑎(3, 3) and 𝑏(−1, 1) on graph paper.

Find the midpoint of [𝑎𝑏].

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2000 FL P2 Q4a

𝑝(5,−8) and 𝑞(11, 10) are two points.

Find the co-ordinates of the midpoint of 𝑝𝑞 .

2004 OL P2 Q2a

Find the co-ordinates of the mid-point of the line segment joining the points (2, − 3) and (6, 9).

2007 OL P2 Q2a

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Solve Problems about Distances Since our axes are perpendicular, we can create a right-

angled triangle from two points:

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Solve Problems about Distances Pythagoras’ Theorem can relate the sides of this triangle

to each other. Knowing the vertical and horizontal side lengths, we can calculate the distance from 𝐴 to 𝐵.

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𝐴𝐵 2 = 42 + 32

𝐴𝐵 2 = 16 + 9

𝐴𝐵 2 = 25

𝐴𝐵 = 5

The distance between two

points 𝐴 and 𝐵is written 𝐴𝐵 .

The distance between points

is also the length of the line segment

between them.

Solve Problems about Distances

If we generalise our points to 𝑃 𝑥1, 𝑦1 and 𝑄 𝑥2, 𝑦2 , we can generalise Pythagoras’ Theorem also:

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𝑃𝑄 2 = 𝑥2 − 𝑥12 + 𝑦2 − 𝑦1

2

𝑃𝑄 = 𝑥2 − 𝑥1 2 + 𝑦2 − 𝑦1 2

e.g. using 𝐴 6,5 and 𝐵 10,8

𝐴𝐵 = 10 − 6 2 + 8− 5 2

𝐴𝐵 = 42 + 32

𝐴𝐵 = 16 + 9

𝐴𝐵 = 25 = 5

This formula is included on pg 18

of the F&T

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Solve Problems about Distances e.g. Find the distance between the following sets of

points:

1. A(5, 0), B(1, 4)

2. C(4, 2), D(7, 6)

3. E(-9, 3), F(7, -7)

4. G(8, 1), H(-2, -5)

5. I(4, -1), J(-5, 9)

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Solve Problems about Distances Find the distance between the two points (3, 2) and (8, 14).

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2003 OL P2 Q2a

Find the distance between the two points (3, 4) and (15, 9).

2005 OL P2 Q2a

𝑝(3, 0) is a point. 𝑡 and 𝑠 are two distinct points on the 𝑦-axis and |𝑝𝑡| = | 𝑝𝑠 | = 5.

Find the co-ordinates of 𝑡 and the co-ordinates of 𝑠.

2009 OL P2 Q2bi

Solve Problems about Slopes The slope of a line is how steep it is.

Bigger slopes are steeper.

Positive slopes go up and right.

Negative slopes go down and right.

Describe each of these slopes as big or small, and positive or negative.

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A B C D

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Solve Problems about Slopes

Formula:𝑚 =Δ𝑦

Δ𝑥=

𝑦2−𝑦1

𝑥2−𝑥1

This formula is sometimes spoken “rise over run”.

e.g. Given points 𝐴(3, 2) and 𝐵(1, 7), the slope from 𝐴 to 𝐵 is:

𝑚𝐴𝐵 =7−2

1−3= −

5

2

What is the slope from 𝐵 to 𝐴?

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Solve Problems about Slopes Find the slope of

each side of the quadrilateral shown:

𝐹 2, 0

𝐺 1, 5

𝐻 3, 7

𝐼(7, 2)

Do any sides have equal slopes?

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𝑚 =𝑦2 − 𝑦1𝑥2 − 𝑥1

Solve Problems about Equations of Lines

Equations of a line are the algebraic representations of lines. They are a description of all of the points that are on the line.

e.g. Given the equation of the line: 2𝑥 + 3𝑦 = 11, a point is on the line only if we substitute the coordinates and the equation is true.

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e.g. 1, 3

⇒ 2 1 + 3 3 = 11

⇒ 2 + 9 = 11

Is true, so (1, 3) is on the line.

e.g. 2,−1

⇒ 2 2 + 3 −1 = 11

⇒ 4 − 3 ≠ 11

So (2, −1) is not on the line.

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To find the equation of a line, we need one point on it and its slope:

Formula: 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1)

Where (𝑥1, 𝑦1) is a point on the line and 𝑚 is its slope.

The 𝑥 and 𝑦 in the formula do not change.

e.g. What is the equation of the line with slope 3 passing through the point (2, 1)?

𝑦 − 1 = 3 𝑥 − 2

⇒ 𝑦 − 1 = 3𝑥 − 6

⇒ 3𝑥 − 𝑦 − 6 + 1 = 0

⇒ 3𝑥 − 𝑦 − 5 = 0

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e.g. Find the equations of the following lines:

a) Passes through (1, 2), slope 3.

b) Passes through (0, 4), slope 2.

c) Passes through (−2, 2), slope -1.

d) Passes through (−1,−3), slope 4.

e) Passes through (4, 2), slope 3

5.

f) Passes through (−2,−1), slope 1

2.

g) Passes through (10, 1), slope 0.

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𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1)


e.g. A plumber charges a €40 callout fee and charges €20 per hour worked.

a) How much would it cost to hire the plumber for 2 hours?

b) Find the equation of the line containing the point (0, 40)and has slope 20.

c) Draw a graph showing the cost of the plumber vs. how

long they work (up to at least 8 hours).

d) Use your graph to find how much the plumber would cost for 5 hours.

e) Use your graph to find how long the plumber worked if they charged €100.

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𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1)

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e.g. Find the equations of the following lines:

a) Passes through 1, 2 and (3, 6).

b) Passes through (0, 4) and (4, 8).

c) Passes through (1, 6) and (3, −3).

d) Passes through (−3, 2) and (2, −3).

e) Passes through (2, −5) and (6, 0).

f) Passes through (−2,−1) and (2, −7).

g) Passes through (0, 3) and (6, 3).

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𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1)

𝑚 =𝑦2 −𝑦1𝑥2 − 𝑥1


e.g. Jessica opens a savings account with €30 in it and commits to depositing €10 more every week.

a) Draw a graph to show how much Jessica saves over 10 weeks. Label your axes appropriately.

b) What is the slope of the graph?

c) Find the equation of the line representing Jessica’s

savings.

d) Use the graph and the equation of the line to find Jessica’s savings after 7 weeks.

e) Use the graph and the equation of the line to find how long it takes Jessica to save €60.

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𝑎(−2, 1) and 𝑏(4, 5) are two points.

a) Plot the points 𝑎 and 𝑏 on a co-ordinate diagram.

b) Find the slope of 𝑎𝑏.

c) Find the equation of 𝑎𝑏.

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2009 OL P2 Q2

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There are two main ways two write equations of lines:

1. 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 or 𝑎𝑥 + 𝑏𝑦 = 𝑐, called standard form.

2. 𝑦 = 𝑚𝑥 + 𝑐, called slope-intercept form.

Which form are the following equations in?

a) 𝑦 = 2𝑥 + 7

b) 2𝑥 + 𝑦 − 6 = 0

c) 5𝑥 + 3𝑦 − 1 = 0

d) 𝑦 =2

3𝑥 − 5

e) −2𝑥 + 1 = 7𝑦

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Exam

questions are

usually

answered

using standard

form.


In standard form (𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0), slopes can be calculated from equations using:

Formula:𝑚 = −𝑎

𝑏

e.g. Find the slopes of the following lines:

a) 3𝑥 − 𝑦 − 5 = 0

b) 2𝑥 + 3𝑦 = 11

c) 5𝑥 − 𝑦 + 2 = 0

d) 3𝑥 = 5𝑦 + 1

e) 0 = 7 − 𝑥 + 2𝑦

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This formula is

not in your

formula and

tables book.


In slope-intercept form (𝑦 = 𝑚𝑥 + 𝑐), the slope can simply be read from the coefficient of 𝑥.

e.g. Find the slopes of the following lines:

a) 𝑦 = 2𝑥 + 5

b) 𝑦 = −𝑥 − 7

c) 𝑦 = 3𝑥 − 2

d) 2𝑦 = 7𝑥 + 1

e) −𝑦 = 3𝑥 + 2

f) 𝑦 = 3

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On a side

note, the line

will cross the 𝑦-

axis at 0, 𝑐 ,

called the 𝑦-

intercept.

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The co-ordinate diagram below shows the lines 𝑞, 𝑟, 𝑠,and 𝑡. 𝑞 is parallel to 𝑠 and 𝑟 is parallel to 𝑡.

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Complete the table.

2017 JC HL P2

Q11


𝐿 is the line 𝑦 − 6 = −2 𝑥 + 1

a) Write down the slope of 𝐿.

b) Verify that 𝑝(1, 2) is on 𝐿.

c) 𝐿 intersects the 𝑦-axis at 𝑡. Find the co-ordinates of 𝑡.

d) Show the line 𝐿 on a co-ordinate diagram.

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2008 OL P2 Q2

Use Slopes to Show that Lines are

Perpendicular or Parallel

Parallel lines have the same slope.

i.e. 𝑚1 = 𝑚2

Perpendicular lines have inverse, negative slopes.

i.e. 𝑚1 ×𝑚2 = −1 ⇒ 𝑚2 = −1

𝑚1

e.g. Given the equation of 𝐿: 2𝑥 + 𝑦 = 4, find:

a) The equation of 𝐾, which passes through (0, 2) and is

parallel to 𝐿.

b) The equation of 𝐽, which passes through (0, 2) and is perpendicular to 𝐿.

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𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1)

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Use Slopes to Show that Lines are

Perpendicular or Parallel

e.g. given lines 𝑙: 𝑦 = 4𝑥 − 7 and 𝑘: 8𝑥 + 𝑐𝑦 = 9

i. Find the slope of 𝑙

ii. Find the slope of 𝑘 in terms of 𝑐.

iii. Find the value of 𝑐 if 𝑙 ∥ 𝑘 (i.e. 𝑙 is parallel to 𝑘).

iv. Find the value of 𝑐 if 𝑙 ⊥ 𝑘. (i.e. 𝑙 is perpendicular to 𝑘).

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Solve Problems about Lines e.g. The line 𝐿 intersects the 𝑥-axis at (−4, 0) and the 𝑦-

axis at (0, 6).

a) Find the slope of 𝐿.

b) Find the equation of 𝐿.

c) The line 𝐾 passes through (0, 0) and is perpendicular to 𝐿. Find the equation of 𝐾.

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2007 OL P2 Q2

e.g. 𝐿 is the line 3𝑥 + 2𝑦 + 𝑐 = 0.

a) (3, −1) is a point on 𝐿. Find the value of 𝑐.

b) The line 𝐾 is parallel to 𝐿 and passes through the point (−2, 5). Find the equation of 𝐾.

2006 OL P2 Q2

Find the Intersection Point of Two Lines

By plotting lines on a coordinate plane, their intersection point can be found.

Find the intersection point of these lines by drawing a graph:

1. 𝑥 + 𝑦 = 4; 𝑥 − 𝑦 = 10

2. 𝑥 + 𝑦 = 6; 2𝑥 − 4𝑦 = 12

3. 5𝑥 − 2𝑦 = 9; 3𝑥 + 𝑦 = 1

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Find the Intersection Point of Two Lines34

Make sure you

draw arrows

from the

intersection to

the axes.

Find the Intersection Point of Two Lines The same point can be found algebraically by solving

the equations simultaneously:

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1. 𝑥 + 𝑦 = 4

2. 𝑥 − 𝑦 = 10

⇒ 2𝑥 = 14

⇒ 𝑥 = 7

1. 7 + 𝑦 = 4

⇒ 𝑦 = −3

- or -

2. 7 − 𝑦 = 10

⇒ −𝑦 = 3

⇒ 𝑦 = −3

Calculate the Area of a Triangle

For a triangle 𝑂𝐴𝐵 with one point, 𝑂, at the origin, 𝐴(𝑥1, 𝑦1) and 𝐵(𝑥2, 𝑦2), the area of the triangle is given by:

Formula: 𝐴𝑟𝑒𝑎 =1

2|𝑥1𝑦2 − 𝑥2𝑦1|

Note the modulus signs, making sure the answer is always positive (since area cannot be negative).

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Calculate the Area of a Triangle e.g. What is the area of triangle OAB if 𝑂 = (0, 0), 𝐴 =(1, 7), and 𝐵 = (5,−2)?

𝐴𝑟𝑒𝑎 =1

21 −2 − 7 5

=1

2−2− 35

=1

2−37 =

37

2

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Calculate the Area of a Triangle If none of the vertices are on the origin, the triangle must

be translated.

e.g. Find the area of triangle ABC if 𝐴 = (3,−1), 𝐵 = (4, 2), and 𝐶 = (−1, 3).

Choose one point to translate to (0, 0) and apply that translation to each point:

𝐴 3,−1−3,+1

𝐴′ 0, 0

𝐵 4, 2−3,+1

𝐵′ 1, 3

𝐶(−1, 3)−3,+1

𝐶′(−4, 4)

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Calculate the Area of a Triangle

𝐴′ = (0, 0), 𝐵′ = (1, 3), 𝐶′ = (−4, 4)

𝐴𝑟𝑒𝑎 =1

2𝑥1𝑦2 − 𝑥2𝑦1

=1

21 4 − 3 −4

=1

24 + 12

=1

216

= 8

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Calculate the Area of a Triangle e.g. Find the area of the triangle with vertices (0, 0), (8, −6), and (−1, 5).

40

2011 OL P2 Q2

2010 OL P2 Q2

2008 OL P2 Q2 e.g. Find the area of the triangle with vertices (0, 0), (8, 6), and (−2, 4).

e.g. 𝐴(−1,−6), 𝐵(6, 8), and 𝐶(2, 5) are three points.

a) Find the area of the triangle 𝐴𝐵𝐶.

b) Find the co-ordinates of the two possible points 𝐷 on the 𝑥-axis such that the area of Δ𝐴𝐵𝐷 = area Δ𝐴𝐵𝐶.

Area of Triangles The line 𝑅𝑆 cuts the x-axis at the point 𝑅 and the 𝑦-axis at

the point 𝑆(0, 10), as shown. The area of the triangle 𝑅𝑂𝑆,

where 𝑂 is the origin, is 125

3.

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a) Find the coordinates of 𝑅.

b) 𝐸(−5, 4) is on the line 𝑅𝑆.

A second line 𝑦 = 𝑚𝑥 + 𝑐passes through 𝐸 and also makes a triangle of

area 125

3with the axes.

Find the values of 𝑚 and 𝑐 if they are both positive.

2014 HL P2 Q5

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