Analytical geometry
Distance
For a vertical line: Ytop – Ybottom
For a horizontal line: Xright – Xleft
The distance formula: d = √(𝑥1 − 𝑥2 )2 + (𝑦1 − 𝑦2 )2
. or d2 = (𝑥1 − 𝑥2 )2 + (𝑦1 − 𝑦2 )2
Midpoints
𝑥1+𝑥2
For an x point: 2
𝑦1+𝑦2
For a y point: 2
𝑥1+𝑥2 𝑦1+𝑦2
The midpoint formula: (xm;ym) = ( ; )
2 2
Vertex to midpoint (medians)
Vertex dropped perpendicular to base (altitude)
Line dropped perpendicular to midpoint (perpendicular bisector)
, The gradient of a line
𝑟𝑖𝑠𝑒 ∆𝑦
M = 𝑟𝑢𝑛 = ∆𝑥
𝑦2−𝑦1
M=
𝑥2−𝑥1
Parallel lines: m1 = m2
Perpendicular lines: m1 x m2 = -1
Points are collinear if the gradient between any two = the gradient between any other 2
Positive gradient
- Makes acute angle with the x-axis
- Function is increasing
- Equation: y = mx + c
Negative gradient
- Makes obtuse angle with the x-axis
- Function is decreasing
- Equation: y = mx + c
Zero gradient
- Parallel to the x-axis
- Constant function
- Equation: y = c
Undefined gradient
- Perpendicular to the x-axis
- Vertical function
- Equation: x = b
The equation of a straight line
Y = mx + c (liner and first degree)
Finding the equation:
1. Get m
2. Substitute into y – y1 = m(x-x1)
3. Simplify
Finding m
∆𝑦
1. Graph ( m = ∆𝑥 )
2. Equation (y = mx+c)
𝑦2−𝑦1
3. 2 points (m = 𝑥2−𝑥1 )
4. Parallel lines (m1 = m2)
5. Perpendicular lines (m1 x m2 = -1)
6. Given angle (m = tan Ɵ)
Distance
For a vertical line: Ytop – Ybottom
For a horizontal line: Xright – Xleft
The distance formula: d = √(𝑥1 − 𝑥2 )2 + (𝑦1 − 𝑦2 )2
. or d2 = (𝑥1 − 𝑥2 )2 + (𝑦1 − 𝑦2 )2
Midpoints
𝑥1+𝑥2
For an x point: 2
𝑦1+𝑦2
For a y point: 2
𝑥1+𝑥2 𝑦1+𝑦2
The midpoint formula: (xm;ym) = ( ; )
2 2
Vertex to midpoint (medians)
Vertex dropped perpendicular to base (altitude)
Line dropped perpendicular to midpoint (perpendicular bisector)
, The gradient of a line
𝑟𝑖𝑠𝑒 ∆𝑦
M = 𝑟𝑢𝑛 = ∆𝑥
𝑦2−𝑦1
M=
𝑥2−𝑥1
Parallel lines: m1 = m2
Perpendicular lines: m1 x m2 = -1
Points are collinear if the gradient between any two = the gradient between any other 2
Positive gradient
- Makes acute angle with the x-axis
- Function is increasing
- Equation: y = mx + c
Negative gradient
- Makes obtuse angle with the x-axis
- Function is decreasing
- Equation: y = mx + c
Zero gradient
- Parallel to the x-axis
- Constant function
- Equation: y = c
Undefined gradient
- Perpendicular to the x-axis
- Vertical function
- Equation: x = b
The equation of a straight line
Y = mx + c (liner and first degree)
Finding the equation:
1. Get m
2. Substitute into y – y1 = m(x-x1)
3. Simplify
Finding m
∆𝑦
1. Graph ( m = ∆𝑥 )
2. Equation (y = mx+c)
𝑦2−𝑦1
3. 2 points (m = 𝑥2−𝑥1 )
4. Parallel lines (m1 = m2)
5. Perpendicular lines (m1 x m2 = -1)
6. Given angle (m = tan Ɵ)
