circle theorems

x = 100

angles at the centre/circumference - the angle at the centre is double that of the angle at the circumference

p = 52, q = 40

angles in the same segment are always equal

cyclic quadrilaterals - opposite angles in a cyclic quadrilateral add up to 180

note - a quadrilateral is only cyclic if all four corners touch the perimeter of the circle

what is angle APB?

APB = 90

angles in a semicircle - angles at the circumference of a semicircle are equal to 90 degrees

what is the angle marked above?

90

tangent theorem 1 - the angle between a tangent and a radius is 90.

if line AP has a length of 8cm, what is the length of line BP?

8cm

tangent theorem 2 - tangents which meet at the same point are equal in length

if the length of the chord from A - B is 5cm, what is the length of the chord from B - C?

5cm

chord theorem - any line going through the centre of a circle which is perpendicular to a chord bisects it, and the angle from any line bisecting a chord and the chord itself is 90 degrees.

what is true about ab and xy?

their products are equal

intersecting chord theorem - When two chords of a circle intersect within the circle, the product of the segments of one chord is equal to the product of the segments of the other chord. (aka ab = xy)

this only applies when they intersect within the circle

what is true about ab and xy?

they are equal

Intersecting secant theorem - When two secants of a circle meet at a point outside of the circle, the product of the external secant segments and the secant segment to the circumference are equal. ( ab = xy)

if one green angle is 80 degrees and one yellow angle is 60 degrees, what is true about the other green and yellow angles?

the other green angle = 80 degrees, the other yellow angle = 60 degrees

alternate segment theorem - The angle between a tangent and a chord is equal to the angle in the alternate segment.