Circle Theorems IGCSE Maths
Summary
TLDRThis educational session delves into the properties of a circle, defining it as a set of points equidistant from a central point. Key concepts covered include the radius, circumference, diameter, and the relationships between them. The session also explores tangents, secants, and chords, emphasizing the perpendicularity of the radius to the tangent and the bisector property of chords. It discusses angles in a semicircle being 90° and the relationship between angles at the center and on the circumference. The video concludes with insights on cyclic quadrilaterals, where opposite angles are supplementary, and the sum of all angles equals 360°.
Takeaways
- 🔵 A circle is defined as a set of points equidistant from a fixed point known as the center.
- 📏 The radius of a circle is the distance from the center to any point on the circumference.
- 🌀 The circumference of a circle is calculated as \( 2\pi r \), where \( r \) is the radius.
- 📏 The diameter of a circle is the longest chord that divides the circle into two equal parts, and it is equal to twice the radius (\( D = 2R \)).
- 📐 A tangent to a circle is a line that touches the circle at exactly one point.
- 📐 A secant is a line that intersects a circle at two points, extending beyond the circle.
- ✂️ The property of equal chords in a circle states that if two chords are equal, their distances to the center are also equal.
- 🔼 The angle in a semicircle is always 90 degrees, reflecting the right angle property of a semicircle.
- 🔼 The angle between a tangent and a radius is always 90 degrees, indicating the perpendicular relationship.
- 🔄 The angle at the center of a circle subtended by an arc is twice the size of the angle on the circumference subtended by the same arc.
Q & A
What is the definition of a circle?
-A circle is the set of all points that are equidistant from a fixed point, known as the center.
What is the term for the distance from the center of a circle to any point on its circumference?
-The distance from the center of a circle to any point on its circumference is called the radius.
How is the circumference of a circle calculated?
-The circumference of a circle is calculated using the formula C = 2πr, where r is the radius of the circle.
What is the term for a line that divides a circle into two equal parts?
-A line that divides a circle into two equal parts is called the diameter, which is also the longest chord in a circle.
What is the relationship between the diameter and the radius of a circle?
-The diameter of a circle is twice the length of its radius, expressed as D = 2R.
What is a tangent in the context of a circle?
-A tangent is a line that touches the circle at exactly one point.
What is a secant in relation to a circle?
-A secant is a line that intersects the circle at two points, extending beyond the circle.
What property ensures that a line drawn from the center of a circle to the midpoint of a chord is perpendicular to the chord?
-The property that ensures a line from the center to the midpoint of a chord is perpendicular is known as the perpendicular bisector theorem.
How are the angles in a semicircle related?
-Any angle inscribed in a semicircle is always a right angle, meaning it measures 90 degrees.
What is the relationship between the angle at the center of a circle and the angle on the circumference subtended by the same arc?
-The angle at the center of a circle subtended by an arc is twice the size of the angle on the circumference subtended by the same arc.
What is a cyclic quadrilateral, and what is one of its properties?
-A cyclic quadrilateral is a quadrilateral whose vertices all lie on a circle. One of its properties is that the sum of the opposite angles is supplementary, meaning they add up to 180 degrees.
Outlines
🌀 Introduction to Circles and Their Properties
This paragraph introduces the concept of a circle and its fundamental properties. A circle is defined as a set of points equidistant from a fixed point known as the center. The distance from the center to any point on the circle is called the radius. The circumference, or perimeter, of a circle is calculated as 2πr, where r is the radius. The paragraph also explains the terms cord and diameter, with the former being any line segment connecting two points on the circle and the latter being the longest possible cord, which passes through the center and divides the circle into two equal halves. The relationship between diameter (D) and radius (r) is given as D = 2r. The properties of tangents and secants are also discussed, with tangents being lines that touch the circle at exactly one point, and secants being lines that intersect the circle at two points. The paragraph concludes with a discussion on the properties of equal chords and the perpendicular bisector of a sector, stating that a line drawn from the center to the midpoint of a chord is perpendicular to the chord, and chords of equal length are equidistant from the center.
📐 Exploring Tangents, Angles, and Cyclic Quadrilaterals
The second paragraph delves deeper into the properties of tangents and angles within a circle. It explains that the length of a tangent from an external point to the circle is constant, leading to the conclusion that two triangles formed by a tangent and a radius are congruent. This congruence implies that angles subtended by the same arc are equal. The paragraph also discusses the angle in a semicircle, which is always 90 degrees, and uses this property to solve for angles in various geometric configurations. The concept of an external angle being equal to the sum of the two non-adjacent interior angles is also explored. The paragraph concludes with an explanation of the properties of angles at the center and on the circumference of a circle, stating that the angle at the center is twice that of the angle on the circumference subtended by the same arc. This knowledge is applied to solve problems involving angles in cyclic quadrilaterals, where the sum of opposite angles is supplementary.
🔄 Understanding Cyclic Quadrilaterals and Angle Properties
The final paragraph focuses on cyclic quadrilaterals and their properties. A cyclic quadrilateral is defined as a quadrilateral whose vertices all lie on a circle. The paragraph discusses two key properties of cyclic quadrilaterals: the sum of all interior angles equals 360 degrees, and the sum of opposite angles is supplementary (i.e., they add up to 180 degrees). Using these properties, the paragraph solves for unknown angles within a cyclic quadrilateral. It also revisits the concept of angles subtended by the same arc being equal, which is used to find the values of angles in different segments of the circle. The paragraph concludes by summarizing the properties of circles and their angles, and hints at upcoming videos that will cover past paper questions related to these topics.
Mindmap
Keywords
💡Circle
💡Center
💡Radius
💡Circumference
💡Cord
💡Diameter
💡Tangent
💡Secant
💡Perpendicular Bisector
💡Angle in a Semicircle
💡Cyclic Quadrilateral
Highlights
Definition of a circle as a set of points equidistant from a fixed point, known as the center.
Explanation of the radius as the distance from the center to the circumference of a circle.
Circumference of a circle is calculated as 2πr, where r is the radius.
Introduction to the concept of a cord in a circle, dividing it into two parts.
Definition of a diameter as the longest cord that divides the circle into two equal parts, being twice the radius.
Tangent is a line that touches the circle at a single point.
Secant is a line that intersects the circle at two points, extending beyond the circle.
Property of equal chords and the perpendicular bisector in a circle, dividing the chord into two equal parts.
The property that any angle in a semicircle is always 90 degrees.
Tangent lines from an external point to a circle are equal in length.
Angle between a tangent and the radius is always 90 degrees.
Angle at the center of a circle is twice the size of the angle on the circumference subtended by the same arc.
Property of angles in the same segment of a circle being equal.
Cyclic quadrilateral is defined by all vertices lying on the circumference of a circle.
Sum of all angles in a cyclic quadrilateral is 360 degrees.
Opposite angles in a cyclic quadrilateral are supplementary.
Practical application of circle properties to solve geometry problems.
Transcripts
in this session we will cover the topic
related to Circle and its
properties hello and welcome to the
session so today we are going to cover
Circle and the circle properties what is
circle circle is the set of point which
is always equid distance from the fixed
point and the fixed point is nothing but
the center okay so center of a circle
okay from the fixed point to all the
point which are equal distance if I will
try to draw a line this line is nothing
but all the distance between these two
point is known as
radius okay so radius is the distance
from Center to the circumference of a
circle okay now this whole length right
this whole length is known as
circumference okay or the perimeter we
can say circumference of a circle so
circumference is nothing but 2 into piun
into R where R is a radius okay now
there is a if you see this is one line
which is dividing the circle in two
parts this line is nothing but the cord
so cord is a line which divide the
circle in two parts okay now if you'll
see a cord which is passing through the
center it is dividing into two equal
parts right so so that cord or the
longest cord is known as
diameter okay so diameter is nothing but
it is a longest cord which divide the
circle in two equal parts and this is
the two times of the radius so if you'll
see r + r which is going to be the
diameter so D is equal 2 R it is the
relation between the diameter and the
radius now tangent tangent is a line
which touches the circle at a single
point if you if I draw a line like this
one if you'll see at this point it is
touching right so it touches the circle
at a single point so we can say this is
nothing but
tangent okay so tangent is a line which
touches the circle at a single point
okay now secant secant is a line it's a
extension of the cord but like like this
that means it is what intersecting at
two point so when it intersect at two
point we call it a secant okay so secant
is nothing but it is an extension of
a Cod but it intersect at two point
instead of touching like if you see it
is inside the circle it is what outside
it is intersecting at this point and
this point so we call it
a second okay now coming to the circle
properties equal cord and the
perpendicular bis sector okay so if
suppose if I'm going to draw a circle
consider this is a circle
okay if I draw a cord okay so any line
which is drawn from
cord to the circle if you'll see this
one right so this line is perpendicular
if this line is perpendicular then it
bisect the cord in two equal part okay
okay also if the circle is like consider
this is a circle I'm considering this
circle okay
now one cord is this one and just take
another cord is this one if cords are
equal
right okay say if cords are
equal then their distance is also same
from the center okay okay so equal cord
and the perpendicular bis sector if
these are the perpendicular bis sector
then it will divide into the two equal
part it will divide into the two equal
part this we call it radius this we call
it a radius okay in this scenario we can
take this as a radius and this as a
radius if you'll see here these two
triangles are congruent right so if
those two triangles are congruent that
means each part like this angle is
equals to this this angle is going to be
equals to this this angle and this angle
is going to be equal so this is one the
cord if a line drawn perpendicular from
cord uh from center of the circle to the
cord it will bisect the cord okay or in
other way we can say if a line which is
bisecting the
cord then it will be perpendicular to
the cord okay now next if you'll see
here tangent we already discussed right
tangent is the line which touches the
circle at a single point now any tangent
which is drawn from external point right
how many triangles we can draw two
triangles maximum we can draw two
triangles if you'll see and this tangent
length of the tangent is always going to
be same that means AC is equals to
BC okay so in this scenario if we'll see
AC and BC is equals to be same OB and O
A is radius so I'm going to write here
radius r
okay this and this is going to be same
so I'm going if it is X then this is
also going to be X right and this is
what a common right if you'll see this
is what common so both the triangles are
congruent if both the triangles are
congruent using rhs right angle rhs is
like right angle hypotenuse and one side
right angle is there if you'll see
hypoten is OC and one side is like OB
and O A we can consider right so in this
scenario both the triangles are
congruent if both the triangles are
congruent in that scenario each and
every part will be equal that means this
going to be equals to this right this
angle is going to be this one okay so if
you'll see we'll use the same property
over here this is 20 so I'm going to
write here this is going to be 20 if
this is X I'm going to write here x
now this is what radius and tangent is
always going to be 90° so this is going
to be 90 90 20 and X this is a right
angle triangle right so we can write
here x + 90 + 20 is = 180 x is going to
be 180 - 110 x = to 70° so like this way
we can identify the value of x angle in
a semicircle is always going to be 90°
if you see this is a Ab is a what cord
which is the longest cord which is
dividing into the two parts so this is
what AB this part is semicircle so angle
any angle if you draw it is always going
to be 90° angle in a semicircle is
always going to be 90° so this is 90
this is going to be 90 C1 and C2 is
going to be same so let's solve this
question if you'll see the first
question this is going to be what I
writing here this is 90° because angle
in a semicircle now this angle we know 9
20 + 45 so this angle we can suppose y
so
X sorry 45 + 90 + y is going to be 180 y
= 180 - 90 + 45
135 subtract it 80 - 5 so this is 5 7 -
3 is 5
55 okay uh 45 okay so 45 my bad so 4 45
so this angle is 45 so I can write here
this is 45 so X we have to identify X is
equal to 18 - 45 why because this is a
linear pair straight line so I can write
here x equal to 135 or we can use
directly property this is what external
angle this is what opposite to external
angle right external angle is equals to
the sum of the Interior opposite angle
that means 90 + 45 which is going to be
135 okay similarly here it is going to
be 90 so this is going to be 20 right
because 1 10 so this is going to be
70 90 70 and 20 like this way we can
write angle between the tangent and the
radius of the circle so angle between
the tangent tangent is always going to
be
90 okay tangent is always going to
perpendicular to the radius that means
this is 90° okay let's solve this one
this is going to be 90° this is going to
be
90° and this is what x so this is what a
quadrilateral x + 90 + 140 + 90 is going
to be 360 x is going to be 360 - 180 -
140 right 9090 180 and - 140 180 - 40
which is going to be 40 or we can
directly
write angle at the center of a circle
so angle at in this if you the angle
subtended at the center of a circle by
an arc this is one Arc which is subing
angle at the center is always twice the
size of the angle on the circumference
subtended by the same Ark so if it is
going to be X then this is going to be
2x if it is going to be 2x then this is
going to be X okay so this is our angle
at the center property so in this case
if you'll see o
this is going to be 40 I'm going to
write why 40 because this is what radius
this is what radius this is an isos
triangle so this angle we can find out
why because angle some property right so
40 40 and this is going to be 100 so
angle I'm writing a z so Z is going to
be 100 now angle at the center by this
Arc if you'll see it is making angle Z
which is 100 so X is going to be half of
this which is going to be
50 okay now we know X now this is what
if you see this and this is equal that
means this angle and this angle is going
to be equal so I am going to write here
x this is going to x y +
40 y +
40 again y + 40
because these two angle will be equal
right if it is 40 40 this is y so this
is also going to be y so y +
40 is equal to 180° so in this scenario
I can write here x 2 * of y + 40 isal to
180 x we know right 50 substitute It 2 y
+ 40 is going to be 180 -
50 y + 40 is going to be 130 / 2 which
is 65 so y = 25 so like this way we can
find out the value of y angle in the
same segment so angle in the same
segment are always equal so if you'll
see this is one Arc which is making
angle X and again this is also making
angle X so those are equal similarly
here if 33 this is what making 33 with
the help of this Arc right so this is B
also going to be 33 so like this way we
can use this property to identify the
value of angle in the same Arc okay now
if you'll see this angle how we can find
this angle and this angle is going to be
equal why because this is what opposite
angles right so if it is X then this is
going to be X right and again if I use
this property angle in the same segment
right
X then this is suppose y so y so this is
going to be Y and Y right so angle some
using this property we can identify the
value of missings okay so like this way
we can fine now if you see cyclic
quadrilateral what is cyclic
quadrilateral if all the point of a
quadrilateral if you see p q r s all the
vertices of a quadrilateral are lying on
the circum of a circle or on the circle
we can say cyclic quadal so what is
property of a cyc quadrilateral First
Property simple quadrilateral if it is
there then sum of all the angle is going
to be 360
PQ r + S equal to
360° another property of the CYCC
quadrilateral is opposite angle sum of
the opposite angle is going to be
supplementary that means p+ angle R is
going to be
180 angle S Plus angle Q is going to be
180 so simple case this is also a
example of cyc quad a + 100 is 180 so a
is going to be 80° and B is going to 180
minus
115 subtract it 65 so like this way we
can identify so this is all about this
circle and the properties of a circle I
will also come with another videos were
from the past paper questions please
share and subscribe thank you for
watching
5.0 / 5 (0 votes)