What is an arc?
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What is an arc?
An arc is a portion of the circumference of a circle.
What is the circle theorem that describes an angle formed at the centre and another at circumference of a circle from lines starting at the same two points on the circumference of the circle?
The circle theorem that describes the angles at the centre and circumference of a circle is: The angle subtended by an arc at the centre is twice the angle at the circumference.
Note that both angles need to share the same arc (it should look like an arrow head not a kite).
If you know the angle at the centre, how can you find the angle at the circumference that is formed by two lines from the same points on the circumference but subtended by the opposite arc?
If you know the angle at the centre, you find the angle at the circumference that is formed by two lines from the same points on the circumference but subtended by the opposite arc by doing the following:
Find the other angle at the centre that is subtended by the opposite arc
Halve this angle to find the angle at the circumference
What is a diameter?
A diameter is a line segment that passes through the centre of a circle and connects two points on the circumference.
What is the circle theorem that describes a triangle that has the diameter of the circle as one of its lengths?
The circle theorem that describes a triangle that has a diameter of the circle as one of its lengths is: The angle in a semicircle is a right angle.
If one edge of the triangle inside a circle is the diameter, then the triangle is contained within half of the circle (a semicircle) and the angle opposite the diameter is 90º.
True or False?
A triangle in a circle, where all three points touch the circumference of the circle, will always be a right-angled triangle.
False.
A triangle in a circle, where all three points touch the circumference of the circle, will not always be a right-angled triangle.
It will only be a right-angled triangle if one of the sides is the diameter of the circle.
True or False?
The angle in a semicircle is a right angle circle theorem is a special case of the angle subtended by an arc at the centre is twice the angle at the circumference circle theorem.
True.
The angle in a semicircle is a right angle circle theorem is a special case of the angle subtended by an arc at the centre is twice the angle at the circumference circle theorem.
If the angle at the centre is 180º, then the two lines forming the angle from points on the circumference form a straight line, the diameter. This means that the angle at the circumference, subtended by the same arc, must be 90º.
What is a chord?
A chord is any straight line that joins two points on the circumference of a circle.
What is a radius?
A radius is a line segment connecting the centre of a circle to a point on the circumference.
Define the term perpendicular bisector.
A perpendicular bisector is a line that cuts another line exactly in half (bisect) and crosses it at a right angle (perpendicular).
True or False?
The perpendicular bisector of a chord passes through the centre of the circle.
True.
The perpendicular bisector of a chord passes through the centre of the circle.
Therefore a radius is a perpendicular bisector of a chord. This is a circle theorem that you need to remember as it can be very useful in finding equal lengths and angles.
What type of triangle is formed in a circle from two radii and a chord?
If a triangle is formed from two radii and a chord within a circle it will be an isosceles triangle as the two radii are two sides of the same length.
What is a tangent?
A tangent is a straight line that touches the circle at exactly one point.
What is the circle theorem that describes the relationship between a radius and a tangent?
The circle theorem that describes the relationship between a radius and a tangent is: A radius and a tangent are perpendicular (meet at a right angle).
If two tangents to a circle intersect, what can be said about the distances on each line between the point of intersection and the circle?
Two tangents from a circle to the same point outside the circle are equal in length between the point at which they meet the circle and the point of intersection.
What geometrical shape is formed by the centre of a circle, two tangents on the same circle that intersect and two radii.
A kite is formed by two tangents on the same circle that intersect and the centre of the circle.
The two radii form two adjacent sides of equal length and the tangents form the other pair of adjacent sides of equal length. The angle between each radius and tangent is 90º.
What is a cyclic quadrilateral?
A cyclic quadrilateral is a quadrilateral inside a circle with all four vertices lying on its circumference.
What is the circle theorem that describes the relationship between angles in a cyclic quadrilateral?
The circle theorem that describes the relationship between angles in a cyclic quadrilateral is: Opposite angles in a cyclic quadrilateral add up to 180º.
True or False?
Opposite angles add up to 180º for a quadrilateral in a circle that has one vertex at the centre of the circle and the other three on the circumference.
False.
For a quadrilateral in a circle that has one vertex at the centre of the circle and the other three on the circumference, opposite angles do not add up to 180º.
All four vertices must lie on the circumference of the circle for it to be a cyclic quadrilateral.
True or False?
Any two angles at the circumference of a circle that are formed from the same two points on the circumference are equal.
True.
Any two angles at the circumference of a circle that are formed from the same two points on the circumference are equal.
The circle theorem states: Angles at the circumference subtended by the same arc are equal or Angles in the same segment are equal.
What is a segment?
A segment is a region bounded by a chord and an arc of a circle.
What is a cyclic triangle?
A cyclic triangle is a triangle within a circle where all three vertices of a triangle lie on the circumference.
True or False?
The alternate segment theorem states that the angle between a chord and a tangent is equal to the angle in the alternate segment.
True.
The alternate segment theorem states that the angle between a chord and a tangent is equal to the angle in the alternate segment.
True or False?
You can spot the alternate segment theorem by just looking for a cyclic triangle.
False.
You cannot spot the alternate segment theorem by just looking for a cyclic triangle. You must also look to make sure that one of the vertices of the cyclic triangle touches the same point on the circumference as a tangent to the circle.
True or False?
For two chords, AB and CD that meet at point P, the ratio of longer lengths (of chords) ≡ ratio of shorter lengths (of chords).
True.
For two chords, AB and CD that meet at point P the ratio of longer lengths (of chords) ≡ ratio of shorter lengths (of chords).
This is known as the intersecting chord theorem.
How is the intersecting chord theorem used to solve problems?
For two intersecting chords, AB and CD that meet at point P, the easiest way to use the intersecting chord theorem is to use it in its multiplicative form,
AP × PB = CP × PD.
The product of the longer and shorter parts of one chord is equal to the product of the shorter and longer parts of the other chord.
What is a secant?
A secant is a line that extends through a circle, intersecting it at two points.
True or False?
The intersecting secant theorem applies when two chords intersect outside the circle.
True.
The intersecting secant theorem is a version of the intersecting chords theorem and applies when two chords intersect outside the circle instead of inside.
How is the intersecting secant theorem used to solve problems?
For two intersecting secants, ABP and CDP that meet at point P, the easiest way to use the intersecting secant theorem in the form
BP(AB + BP) = DP(CD + DP)