GCSE or General Certificate of Secondary Education is an academic qualification. This course includes numerous subjects pursued by the pupils of secondary education in England, Wales and Northern Ireland. All pupils are required to study English, Mathematics and Science along with various subjects.
Circle theoremsare a number
of theorems related to the circle. It is a significant component of the GCSE
curriculum. Let’s discuss in details the intricacies of circle theorems.
Terms You Should Know To Understand Circle Theorems
You need to understand some circle rules, terms and equations to solve problems on different circle theorems:
‘radius’ of a circle is a term used to explain the distance from the centre to
the edge of the circle.
‘diameter’ is used to describe the distance between one of the edges through
the centre to the other edge of the circle. The diameter is twice the value its
radius of the circle.
‘circumference’ is the distance around the edge of a circle.
‘tangent’ to a circle is a line that touches the circle at one point without
cutting across the circle. The angle between a tangent and radius of the circle
is 90°. The tangent
positioned outside the circle should be equal in length.
‘chord’ is a line that goes from one point to another on the circumference of
‘arc’ is a part of the circumference of the circle.
‘sector’ is a portion of a circle.
‘segment’ explains the portion of a circle between the chord and the associated
arc. The segment is of two types: minor segment and major segment.
Now, that you know all the basic terms, let’s understand circle theorems
An Insight into the Types of Circle Theorems
Circle theorems include several theorems. Some fundamental theorems
included in the GCSE curriculum are:
Milne-Thomson circle theorem of fluid dynamics
Let’s discuss each theorem in detail.
- Inscribed Circle Angle Theorems
It is one of the significant theorems among the circle angle theorems. The theorem is related to the measurement of
an inscribed angle to that of the central angle, subtending the same arc.
Image 1: Diagram of Inscribed Angle Theorem
The value of an inscribed angle is equal to one-half of the value of its intercepted arc. Generally, the proof begins with the case when one side of the inscribed angle is a diameter. Then the angles inside a circle is an external angle of an isosceles triangle. In situations where a side of the inscribed angle is not a diameter, it can be reduced to the former by appropriate auxiliary lines.
Thales theorem is a special case of the theorem of inscribed angle. It was
proved in the 31st proportion of the third book of Euclid’s
Image 2: Diagram of Thales theorem
The Thales theorem is used to find the centre of the circle. In the
figure, a right angle whose vertex is situated on the circle always cuts off
the diameter of the circle. The points P and Q are always at the end of a
diameter line. On drawing two diameters, the centre is found at the point where
the diameters intersect.
- Alternate Segment Theorem
Alternate segment theorem helps in finding angles in circle.
Image 3: Diagram of Alternate segment theorem
The alternate segment theorem states that an angle between a tangent and
a chord through a point of contact is equal to the angle in the alternate
The theorem is named after the Greek mathematician and astronomer
Ptolemy. Ptolemy used the theorem to create his table of chords, a
trigonometric table that was applied to astronomy.
Image 4: Ptolemy’s theorem
Ptolemy’s theorem states that if a convex quadrilateral ABCD is
inscribed in a circle, the sum of the products of two pairs of opposite sides , is equal to
the product of two diagonals.
- Milne-Thomson Circle Theorem
Milne-Thomson circle theorem is a statement that provides a new stream
function for fluid flow.
Image 5: A generalised image of the Milne-Thomson theorem
Let w = f (z) be the complex stream function for fluid flow with no rigid boundaries and no singularities within |z| = a. If a circular cylinder |z| = a is placed in that flow, the complex potential for the new flow is given by:
Image 6: Formula
Image 6: Five circles theorem diagram
The five circles theorem states that if five circles are centred on a
common sixth circle, and it intersects each other chain wise on the same
circle, then the lines joining to the second intersection points to form a
Image 7: Diagram of the six circles theorem
The six circles theorem states that in a chain of six circles together
with a triangle, each circle lies tangent to the two sides of the triangle. It
should also precede the circle in the chain. The chain should close in a way
that the sixth circle is always tangent to the first circle.
The tangent circle theorem includes two theorems:
A radius is obtained by joining the centre and the tangency point. The
tangent at a point on the circle lies at a right angle to the radius.
Image 8: Diagram of Theorem 1
The theorem states that if one draw’s two tangents from an external
point to a circle, then they have two equal tangents. Tangent segment implies
that a line should join the external point to the tangency point. Here, AC= BC.
Take a look at the following diagram:
Image 9: Theorem of Diagram 2
Now, you may have an idea of some of the basic circle theorem included
in the GCSE curriculum.
Scoring high grades in circle theorem largely depend on the ability of a
student to recognise and remember various angle rules and circle formulae. Understand
circle theorems and rules to solve your GCSE assignments better. With regular
practice, you will soon be able to handle circle theorem questions with ease
and fetch excellent grades.
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