co-ordinate geometry 1 contents 1.distance between points (simple)distance between points (simple)...
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1. Distance between points (Simple)2. Pythagoras and Distance between two
points3. The Distance Formula4. Midpoint Formula5. Gradient6. Equations of straight line7. Parallel Lines
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y10.1 Distance between Two Points
(1/5)
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55
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-1-1-2-2
-3-3
-1-1
Distance from (0,3)to (7,3) = 7
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y10.1 Distance between Two Points
(2/5)
11 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414 15150011
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-1-1-2-2
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Distance from (7,-3)to (7,3)= 6
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y10.1 Distance between Two Points
(3/5)
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-1-1-2-2
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Distance from (2,5)to (7,5) = 5
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y10.1 Distance between Two Points
(4/5)
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Distance from (13,7)to (13,2)= 5
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y10.1 Distance between Two Points
(5/5)
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Distance from (3,-2)to (12,-2)= 9
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11 22 33 44 55 66 77 880011
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10.1 Distance – Pythagoras (1/3)
Distance from (1,1)to (4,5)= 5
33
44
c2 = a2 + b2
c2 = 32 + 42
c2 = 9 + 16c2 = 25c = 5
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11 22 33 44 55 66 77 880011
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10.1 Distance – Pythagoras (2/3)
Distance from (2,2)to (7,6)= 41
55
44c2 = a2 + b2
c2 = 52 + 42
c2 = 25 + 16c2 = 41c = 41
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10.1 Distance – Pythagoras (3/3)
Distance from (-1,-2)to (5,5) = 85
66
77
c2 = a2 + b2
c2 = 62 + 72
c2 = 36 + 49c2 = 85c = 85
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10.1 Distance – Pythagoras (1/1)
Distance from (1,-2)to (7,7)
66
99
c2 = a2 + b2
c2 = 62 + 92
c2 = 36 + 81c2 = 117c = 10.816 65c ≈ 10.8
≈ 10.8
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10.2 Distance Formula (1/3)
Distance from (3,-2)to (7,5)?
(3,-2)
(7,5)
d = (x2-x1)2+(y2-y1)2
= (7 - 3)2+(5 --2)2
(x(x11,y,y11))
(x(x22,y,y22))
= 42 + 72
= 16 + 49
= 65(as surd)(as surd)
ExactExact≈ 8.06 (approx)(approx)
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y10.2 Distance Formula (2/3)
Distance from (12,2)to (4,5)?
d = (x2-x1)2+(y2-y1)2
= (4 - 12)2+(5 -2)2
(x(x11,y,y11)) (x(x22,y,y22
))
= (-8)2 + 32
= 64 + 9
= 73(as surd)(as surd)
ExactExact
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y10.2 Distance Formula (3/3)
Distance from (6,2) to (7,5)?
d = (x2-x1)2+(y2-y1)2
= (7 - 6)2+(5 -2)2
(x(x11,y,y11)) (x(x22,y,y22
))
= 12 + 32
= 1 + 9
= 10(2 decimal places)(2 decimal places)
ApproximateApproximate= 3.16
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10.3 Midpoint Formula (1/2)
Midpoint of (3,-2)to (7,6)?
(3,-2)
(7,6)
(x(x11,y,y11))
(x(x22,y,y22))
Midpoint
MM..M =(x1+x2
2y1+y2
2 ),=(3+7
2-2+6
2 ),= (5, 2)
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y10.3 Midpoint Formula (2/2)
Midpoint of (12,2)to (4,5)?(x(x11,y,y11)) (x(x22,y,y22
))
M =(x1+x2
2y1+y2
2 ),=(12+4
22+5
2 ),= (8, 3.5)
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44
7-37-3
10.4 Gradient (1/5)
GradientGradientis the rate of change.rate of change.
Gradient =Horizontal Run
Vertical Rise
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(1,3)(1,3)
(5,7)(5,7)
5-15-1
44m m = rise
run
= 44
= 1
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10.4 Gradient (2/5)
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(5,1)(5,1)
(1,7)(1,7)
1-51-5
66
m = riserun
= 6-4
= 3-2
= -32
WolframWolframDemoDemo
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y10.4 Gradient (3/5)
GradientsGradients can be:
PositivePositive An Increasing Increasing function
ZeroZero HorizontalHorizontal
NegativeNegative A Decreasing Decreasing function
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y10.4 Gradient Formula (4/5)
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yy
yy
(x(x22,y,y22))
(x(x11,y,y11))
yy22-y-y11
xx22-x-x11
m =riserun
=y2-y1
x2-x1
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y10.4 Gradient Formula (5/5)
GradientGradient of (4,2)to (8,10)?(x(x11,y,y11)) (x(x22,y,y22
))
m =y2-y1
x2-x1=
10-28-4
= 84
= 2
GradientGradient of (5,9) to (7,3)?(x(x11,y,y11)) (x(x22,y,y22
))
m =y2-y1
x2-x1=
3-97-5
= -62
= -3
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y10.5 Linear Equations
Two types of equation.
1. Gradient-Intercept Form1. Gradient-Intercept Form
2. General Form2. General Form
y = mx y = mx +b+b
GradienGradientt
y-Intercepty-Intercept
ax +by + c = ax +by + c = 00
‘‘a’ a’ always always
positive.positive.
‘‘b’b’ NOT NOTy-y-
interceptintercept
Always ‘0’Always ‘0’
We must be able to convert between We must be able to convert between forms.forms.
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y10.5 Linear Equations (1/6)
Write in General FormWrite in General Form
a) y = 2x + a) y = 2x + 33
-y-y -y-y0 = 2x – y + 30 = 2x – y + 32x –y +3 = 02x –y +3 = 0
b) 2y = x +7b) 2y = x +7-2y-2y -2y-2y
0 = x – 2y +70 = x – 2y +7x – 2y +7 = 0x – 2y +7 = 0
c) y = -x + 6c) y = -x + 6+x -6+x -6 +x -6+x -6
x + y - 6 = 0x + y - 6 = 0
d) y = +3d) y = +3x3x3
x3x3
xx33
x3x3
3y = x + 93y = x + 9-3y-3y -3y-3y
0 = x -3y + 90 = x -3y + 9x -3y + 9 = 0x -3y + 9 = 0
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y10.5 Linear Equations (2/6)
Write in Gradient-Intercept FormWrite in Gradient-Intercept Form
a) 3y = 9x + 6a) 3y = 9x + 6÷÷33
÷÷33
÷÷33y = 3x + y = 3x +
22
b) 5y = 2x + 1b) 5y = 2x + 1÷÷55
÷÷55
÷÷55y = x +y = x +22
551155
c) y - 2x = 0c) y - 2x = 0+2x+2x +2+2
xxy = 2xy = 2x
d) 3y + x -1 = d) 3y + x -1 = 00 -x-x -x-x+1+1 +1+1
3y = -x +1 3y = -x +1
y = x +y = x +-1-133
1133
÷3÷3 ÷3÷3 ÷3÷3
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y10.5 Linear Equations (3/6)
Write the Gradient and Y-InterceptWrite the Gradient and Y-Intercept
a) y = 9x + 6a) y = 9x + 6 b) y = x - 1b) y = x - 1
c) y = c) y = 2x2x
d) y = x + 7d) y = x + 7
1122
m = 9m = 9
b = 6b = 6m = m =
b = -1b = -1
1122
m = 2m = 2
b = 0b = 0m = 1m = 1
b = 7b = 7
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y10.5 Linear Equations (4/6)
Write the equation. Write the equation. y = mx + by = mx + b
a) m=2 b=1a) m=2 b=1 b) m= b=-5b) m= b=-5
c) m=-4 b=-c) m=-4 b=-11
d) m=12 b=7d) m=12 b=7
2233
y = 2x + 1y = 2x + 1 y = x - 5y = x - 52233
y = -4x - 1y = -4x - 1 y = 12x + y = 12x + 77
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y = x + y = x +
10.5 Linear Equations (5/6)
Write the equation as Write the equation as y=mx+by=mx+b and state and state mm and and bb..
a) y – 2x = 1a) y – 2x = 1 b) 2y = 3x + 7b) 2y = 3x + 7
3322
+2+2xx
+2+2xxy = 2x + y = 2x +
11m = 2m = 2
b = 1b = 1
÷2÷2÷2÷2 ÷2÷2
7722
m =m =
b =b =
33227722
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y10.5 Linear Equations (6/6)
Is the point given on the line?Is the point given on the line?
a) y = 2x + 1 a) y = 2x + 1 (2,5)(2,5)
b) 2y = 3x + 7 b) 2y = 3x + 7 (1,4)(1,4)
c)c) y – 2x = 1 y – 2x = 1 (1,3)(1,3) d) d) y = 5x + 1 y = 5x + 1 (1,5)(1,5)
55 = 2 = 2xx22 + 1 + 1
5 = 55 = 5 YesYes22x44 = 3 = 3x11 + 7 + 7
8 = 8 = 1010
NoNo
33 - 2 - 2xx11 = 1 = 1
1 = 11 = 1 YesYes55 = 5 = 5xx11 + 1 + 1
5 = 65 = 6 NoNo
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y10.6 Parallel Lines (1/5)
Are two lines parallel?Are two lines parallel?
Do they have the same gradient?
mm11 = = mm22To find out if two lines are parallel
put in the form yy == mxmx ++ bb
Check to see if coefficients of x Check to see if coefficients of x are equal.are equal.
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y10.6 Parallel Lines (2/5)
Are the two lines parallel?Are the two lines parallel?
10x + 2y - 7 = 10x + 2y - 7 = 00
5x + y - 3 = 5x + y - 3 = 002y - 7 = -10x
2y = -10x + 7y = -5x + 3.5
y - 3 = -5xy = -5x + 3
mm11 == -5 -5 mm22 == -5 -5
They are parallel !They are parallel !
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y10.6 Parallel Lines (3/5)
Are the two lines parallel?Are the two lines parallel?
6x + 2y - 3 = 06x + 2y - 3 = 0 3x + y - 3 = 03x + y - 3 = 0
2y - 3 = -6x2y = -6x + 3y = -2x + 1.5
y - 3 = -3xy = -3x + 3
mm11 == -2 -2 mm22 == -3 -3
They are They are NOTNOT parallel !parallel !
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y10.6 Parallel Lines (4/5)
Are the interval and line Are the interval and line parallel?parallel?(2,3) to (5,9)(2,3) to (5,9) 4x + 2y - 6 = 04x + 2y - 6 = 0
m =
y2 – y1
x2 - x1
=
2y - 6 = -4x
y = -2x + 1.5
mm11 == 2 2 mm22 == -2 -2
They are They are NOTNOT parallel !parallel !
4y = -8x + 69 – 35 - 2
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y10.6 Parallel Lines (5/5)
Are the interval and line Are the interval and line parallel?parallel?(2,9) to (5,3)(2,9) to (5,3) 6x + 3y - 5 = 06x + 3y - 5 = 0
m =
y2 – y1
x2-x1
=
3y - 5 = -6x
y = -2x + 1.25
mm11 == -2 -2 mm22 == -2 -2
They are parallel !They are parallel !
4y = -8x + 53 – 95 - 2