yr1 chap 6 circles

• to find the midpoint of a line when you know both the endpoints, find the mean of the x coordinates and the mean of the y coordinates of the endpoints, that is the coordinates of the midpoint
• to find the midpoint of a line when you know both the endpoints, find the mean of the x coordinates and the mean of the y coordinates of the endpoints, that is the coordinates of the midpoint
• a perpendicular bisector passes through the midpoint of a straight line at 90 degrees
• the equation of a circle with centre (0,0) and radius r is:

  x^2 + y^2 = r^2
• the equation of a circle with centre (a,b) and radius r is:

  (x-a)^2 + (y-b)^2 = r^2
• the equation of a circle is derived from pythagoras' theorem
• a straight line can intersect a circle 0, 1 or 2 times
• to find the coordinates of the point where a line meets a circle, rearrange the equations of the line and circle into the form y = ..., then make them equivalent to each other and solve
• a tangent to a circle will be perpendicular to the radius at the point of intersection, so has the negative reciprocal gradient
• the perpendicular bisector of a chord must go through the centre of a circle
• a circumcircle is the unique circle drawn around the three corners of any triangle, its centre is a circumcentre, to find the circumcentre find where the three perpendicular bisectors intersect
• when finding the equation of a circle with a known centre, remember to reverse the sign of the x and y coordinate
• to prove a line is the diameter of a circle, find the midpoint of the line, if this is the centre of the circle then it must be a diameter