Circle Geometry Flashcards
What are theorems one a and b? Give reasons for each.
A line drawn from the centre of a circle perpendicular to a chord bisects that chord.
Reason you must give: line from centre perpendicular to chord.
In the same logic, the line segment joining the centre of a circle to the midpoint of a chord is perpendicular to the chord.
Reason: Line from centre to midpoint of chord.
What is theorem two? Give the reason that comes with it.
The angle that an arc subtends at the centre of the circle is twice the angle it subtends at any point on the circumference.
Reason: Angle at centre = 2x angle at circumference
What are theorems three a and b? Give the reasons that comes with them.
The angle subtended by a diameter at the circumference of a circle is a right angle.
Reason: Angle in a semi circle.
And vice versa, if a chord subtends an angle of 90 degrees at the circumference of a circle, then that chord is a diameter of the circle.
Reason: Converse angle in a semi circle.
What are theorems four a and b? Give the reasons that comes with them.
Angles subtended by a chord (or arc) at the circumference of a circle on the same side of
the chord are equal.
OR Angles in the same segment are equal.
Reason: Angles in the same segment
Use (b) to prove that a quad is cyclic: If a line segment joining two points subtends equal angles at two other points on the same side of the line segment, then these four points are on the circumference of a circle (i.e. they are concylic)
Reason: Converse angles in same segment
What are the corollaries of theorems four a and b? Give the reasons that comes with them.
- Equal chords (or arcs) of a circle will
subtend equal angles at the circumference.
Reason: = chords: =angles - Equal chords (or arcs) of a circle will subtend equal angles at the centre of a circle.
Reason: = chords: =angles - Chords (or arcs) are equal if they subtend equal angles at the circumference or at the
centre of a circle.
Reason: = angles: =chords - The angles subtended by chords (or arcs) of equal length in two difference circles with equal radii, are equal.
Reason: = circles: = chords: = angles
What are theorems five a and b? Give the reasons that comes with them.
The opposite angles of a cyclic quad add to 180 degrees.
Reason: opposite angles of a cyclic quad
If the opposite angels of a quad are supplementary, then the quad is cyclic.
Reason: Converse opposite angles of a cyclic quad
(Use to prove that a quad is cyclic)
What are theorems six a and b? Give the reasons that comes with them.
An exterior angle of a cyclic quad is equal to the interior opposite angle.
Reason: Exterior angle of cyclic quad
If an exterior angle of a quad is equal to the interior opposite angle, then the quad is cyclic.
Reason: Converse exterior angle of cyclic quad
(Use to prove that a quad is cyclic).
What are theorems 7 a and b? Give the reasons that comes with them.
A tangent to a circle is perpendicular to the radius at its point of contact.
Reason: tangent
perpendicular to radius
A line drawn perpendicular to a radius at the point where the radius meets the circle is a tangent to the circle. (Used to prove a line is a tangent)
Reason: Converse tangent is perpendicular to radius
What is theorem 8? Give the reason that comes with it.
Two tangents drawn from the same point outside the circle are equal in length.
Reason: Tangents from the same point
What are theorems 9 a and b? Give the reasons that comes with them.
The angle between a tangent to a circle and a chord drawn from the point of contact is
equal to the angle in the alternate segment.
Reason: Tan-chord theorem
If a line is drawn through the end point of a chord, making an angle equal to an angle in the alternate segment, then the line is a tangent. Used to prove that a line is a tangent to a circle.
Reason: Converse tan-chord theorem.
How do you prove theorem one a?
Given: Circle with centre O and OH perpendicular to AB.
RTP: AM=BM
Construction: radii AO and OB
Proof:
In triangle AMO and BMO
1. OA=OB (radii)
2. Angle M1= Angle M2 (given)
3. OM = OM (common)
Therefore triangle AMO = triangle BMO (RHS)
Therefore AM = BM (Triangle AMO = triangle BMO)
How do you prove theorem two a?
How do you prove theorem five a?
How do you prove theorem 9 a?
