Studied by 2 peoplePerpendicular bisector of chord passes through centre.
Angle between tangent and radius is 90o.
Tangents from same point are equal in length.
Angle in semi-circle is 90o.
Angle at centre is twice angle at circumference.
Angles in same segment are equal.
Opposite angles of cyclic quad add up to 180o.
Alternate Segment Theorem
Same segment theorem
Angles in the same segment are equal
Angle at circumference and centre theorem
The angle at the centre is twice the angle at the circumference
Angles in semi circle theorem
The angle in a semi circle is a right angle
Cyclic quadrilateral theorem
Opposite angles of a cyclic quadrilateral add up to 180*
Tangents from a point theorem
Tangents from a point are equal
Tangents and radius theorem
A tangent meets the radius at a right angle
Bisecting tangent theorem
The line from a point to the centre of a circle bisects the angle between the tangents from a point
Radius and chord theorem
A radius bisects a chord at 90*
Alternate angles
Alternate angled are equal.
parts of a circle
sector - pizza slice segment - wonky semi-circle arc - crust circumference - perimeter chord - dodgy diameter radius - to middle diameter - side to side through middle tangent - straight line outside circle
prove the circle theorem: The angle at the centre is twice the angle at the circumference
w = 180 - 2x z = 180 - 2y w + z + a = 360 (180 - 2x) + (180 - 2y) + b = 360 360 - 2x - 2y + b = 360 2x - 2y + b = 0 b = 2x + 2y
angle at circumference = x + y angle at centre = 2x + 2y
prove the circle theorem: Angles in the same segment are equal
draw 2 radius. prove using circle theorem 1 (angle at the centre is twice the angle at the circumference)
prove the circle theorem: Angle in semi-circle is 90o.
using circle theorem 1 (angle at the circumference is twice the angle at the circumference)
angle at centre is 180 so half is 90
prove circle theorem: tangents to a circle which meet at a point are equal in length
draw 2 radius to the tangent to create 2 semi-circles then draw a line from the centre of circle to the point where the tangents meet.
this creates 2 congruent triangles according to ASS. (side in common, radius edge, 90 degree angle)
important point to note about angles in the same segment.
they must touch the circumference
prove the circle theorem: angles in alternate segments are equal
draw a diameter that is 90 degrees to the tangent. Draw a triangle within the semi circle.
This means that angle in semi circle is 90 and the angle between the tangent and diameter in the alternate segment is also 90.
prove the circle theorem: opposite angles in a cyclical quadrilateral total 180
draw 2 radii.
the 2 angles at the circumference are x and y.
then apply the rule (angle at the circumference is half the angle at the centre).
this means the angles at the centre are 2x and 2y.
2x + 2y = 360 x + y = 180
important note about alternate segment theory
all corners of triangle must be touching the circumference.
The centre of each circle is the origin
Find the equation of the tangent to the circle x^2 + y^2 = 169 at the point B (5, -12)
Find the equation of the tangent to the circle x^2 + y^2 = 225 at the point C (9,12)
Find the equation of the tangent to the circle x^2 + y^2 = 100 at the point D (-8,6)
Find the equation of the tangent to the circle x^2 + y^2 = 289 at the point E (-8,-15)
(use coordinates to find gradient of radius)
(gradient of tangent is the negative reciprocal as it is perpendicular)
substitute y and x in the straight line equation using the coordinates to find c
a circle has a centre (2,5) the point A (11,8)) lies on the circumference of the circle. Find the equation of the tangent to the circle at A.
y = 3x + 41
Note
Flashcard