Sketching Straight Lines
Using x & y intercepts | C cannot be equal to 0 |
Gradient - intercept method |
ax + by = c --> e.g 2x - 5y = 10 |
y = mx + c |
y = 0 when solving x |
Horizontal Line |
2x - 0 = 10 |
gradient = 0 |
x = 5 |
y = c |
x = 0 when solving y |
Vertical Line |
2(0) - 5y = 10 |
gradient = undefined |
-5y = 10 |
x = a |
y = 10/-5 |
a --> x-intercept |
y = -2 |
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Finding the equation of a straight line
Gradient + y-intercept |
Gradient + a point (x1, y1) |
2 points (x1, y1) (x2, y2) |
Distance between 2 points |
Sub into equation |
Sub into equation |
1st find m |
Sub into equation |
y = mx + c |
y - y1 = m(x - x1) |
m = y2 - y1/x2 - x1 |
AB = √(x2 - x1)² + (y2 - y1)² |
Example |
Example |
2nd sub into equation |
Example |
c = 5 | m = -1 |
m = 2 | (2, -0.5) |
y - y1 = m(x - x1) |
A = (3, -5) | B = (-2, 1) |
y = -1x + 5 |
y + 0.5 = 2(x - 2) |
Example |
AB = √(-2 - 3)² + (1 - -5)² |
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y + 0.5 = 2x - 4 |
(-1, 4) , (5, 2) |
AB = √(-5)² + 6² |
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y = 2x - 4.5 |
m = 2 - 4/3 - (-1) |
AB = √25 + 36 |
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m = -2/4 --> -1/2 |
AB = √61 |
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y - 4 = -1/2(x - - 1) |
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y - 4 = - 1/2x - 1/2 |
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y = - 1/2x + 7/2 |
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Middle point of a line
Middle point of a line |
Paralell Lines |
Perpendicular Lines |
Sub into equation - i.e / means divide |
Sub into equation |
Sub into equation |
M = (x1 + x2/2, y1 + y2/2) |
m1 = m2 |
m1 x m2 = -1 or m2 = 1/m1 |
Example |
Example - 1st find m (gradient) |
Example - 1st find m (gradient) |
(3, 5) | (2, 7) |
A (4, 13) | B (2, 9) |
A (-4, 9) | B (2, -6) |
M = 3 + 2/2, 5 + 7/2 |
mAB = 9 - 13/2 - 4 |
mAB = -6 - 9/2 - (-4) |
M = (5/2, 12/2) |
mAB = -4/-2 |
mAB = -15/6 |
M = (5/2, 6) |
mAB = 2 |
mAB = -5/2 |
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C (0, -10) | D (15, 0) |
C (-5, 8) | D (10, 14) |
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mCD = 0 - (-10)/15 - 0 |
mCD = 14 - 8/10 - (-5) |
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mCD = 10/15 |
mCD = 6/15 |
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mCD = 2/3 |
mCD = 2/5 |
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∵ mAB ≠ mCB |
2nd | sub into m1 x m2 = -1 |
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∴ not parralell |
-5/2 x 2/5 = -1 |
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-10/10 = -1 |
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∴ AB ⊥ CD |
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