# Circle Theorems - St Sampsons High School

Circle Theorems A Circle features. Circumference Diameter Radius from the distance across around centre passing of the circle the circle,

Circle to any point on the of through the centre circumference the its circle PERIMETER A Circle features.

Major Segment Ch or d AR C Minor Segment

part joining the touches two a lineofwhich circumference points on the of a at the circumference circle circumference. one

point only chord divides circleto From Italian tangere, into two segments touch Tan g

ent Properties of circles When angles, triangles and quadrilaterals are constructed in a circle, the angles have certain properties We are going to look at 4 such properties before trying out some questions together An ANGLE on a chord An Alternatively

that Angles sits on a Weangle say Angles chord subtended does by notan arc as achange chord the in the

APEX same moves segment around the are circumference equal as long as it stays in the same segment From now on, we will only consider the CHORD, not the ARC

Typical examples Find a and b Very angles often, the exam tries to confuse you by drawing the chords Imagine in the

Chord Angle have a = 44 to see the YOU Angles on the same chord for yourself Imagine the Chord Angle b = 28 Angle at the centre A

Consider the two angles which stand on this same chord What do you notice about the angle at the circumference? Chord It is half the angle at the centre We say If two angles stand on the same chord,

then the angle at the centre is twice the angle at the circumference Angle at the centre Its still true when we move The apex, A, around the circumference 272 A 136

As long as it stays in the same segment Of course, the reflex angle at the centre is twice the angle at circumference too!! We say If two angles stand on the same chord, then the angle at the centre is twice the angle at the circumference Angle at Centre A Special Case

a When the angle stands on the diameter, what is the size of angle a? The diameter is a straight line so the angle at the centre is 180 Angle a = 90 We say The angle in a semi-circle is a Right Angle A Cyclic Quadrilateral

is a Quadrilateral whose vertices lie on the circumference of a circle Opposite angles in a Cyclic Quadrilateral Add up to 180 They are supplementary We say Opposite angles in a cyclic quadrilateral add up to 180 Questions

Could you define a rule for this situation? Tangents When a tangent to a circle is drawn, the angles inside & outside the circle have several properties. 1. Tangent & Radius A tangent is perpendicular to the radius of a circle

2. Two tangents from a point outside circle Tangents are equal PA = PB PO bisects angle APB

Angle between tangent & chord We say The angle between a tangent and a chord is equal to any Angle in the alternate (opposite) segment

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