Y12 Geometry

Length of line

Midpoint 

Gradient

Equation of a line through (x_1, y₁) with gradient m 

....................y - y₁ = m(x - x₁) 

• Two lines are parallel if they have the same gradient. 
• Two lines are perpendicular if one is at a right angle to the other (gradient is negative reciprocal). 
  
  
  
• A perpendicular bisector is a line perpendicular to AND passing through the midpoint of another line. 

Equation of a Circle 

centre (0,0)       x² + y² = r² 

centre (a,b)       (x - a)² + (y - b)² = r²

e.g. show that the equation x² + y² + 4x - 6y - 3 = 0 represents a circle, find its centre & radius (completing the square).
(x + 2)² - 2² + (y - 3)² - 3² - 3 = 0       (x + 2)² + (y - 3)² = 16      C(-2, 3)      r = 4

Circle Theorems 

• The angle in a semi-circle is a right angle.
• The perpendicular bisector of a chord is the diameter. 
•  A tangent meets a radius at a right angle. 

• The gradient of the normal is the same as the gradient of the radius. 
• The gradient of the tangent is the negative reciprocal to the gradient of the normal.