EUCLIDEAN GEOMETRY
- Kabelo Nchabeleng
Euclidean geometry is a mathematical system created by the mathematician -
Euclid.
Euclidean geometry is an axiomatic system, in other words - it comprises statements
that are accompanied with reasons and proofs.
Euclidean geometry is one of the most foundational parts of mathematics that fit into
a deductive and logical system.
In high school, Euclidean geometry is limited to plane (two dimensional) geometry,
but in University, it gets merged with other disciplines including differential calculus,
topology and solid geometry of three dimensions.
Geometry is used in art, architecture and engineering (along with analytical
geometry) and is used to design structures. It is also used in geography when
drawing maps of the Earth (which requires elliptic geometry).
Anyway, enough with the trivia. Let’s get into it.
1. Circle Geometry
There are 9 theorems taught in grade 11 that you need to know how to prove and
how they work. We will be discussing each theorem below.
But before we do, let’s define a few basic circle terminology:
- Radius: This is the line drawn from the centre of a circle to any point on the
circumference of the circle. (plural: radii)
- Diameter: This is a line passing through the centre of the circle which is twice
the length of the radius.
- Chord: This is a line connecting two points on the circumference of a circle,
i.e a line segment with endpoints on the circumference.
- Tangent: This is the line that touches/intersects the circle at only one point.
- Secant: This is a line passing through two points on a circle.
- Arc: This is a part of a circumference of a circle.
- Segment: The area between the arc of a circle and its chord.
- Sector: This is the area between two radii and an arc of a circle.
- Disk: The area of the whole circle
, 1.1 THEOREM 1(A)
Theorem 1A states that a line drawn from the centre of a circle to the midpoint of a
chord is perpendicular to the chord.
In essence, according to the sketch, if AM = MB then OM ⟂ AB which means ∠M 1 =
∠M2 = 90°
Given: Circle with centre O. M is drawn on chord AB such that it bisects AB, i.e. AM
= MB.
RTP: That OM is perpendicular to AB.
Proof:
Draw radii OA and OB to construct triangles OMA and OMB
In triangles OMA and OMB:
- Kabelo Nchabeleng
Euclidean geometry is a mathematical system created by the mathematician -
Euclid.
Euclidean geometry is an axiomatic system, in other words - it comprises statements
that are accompanied with reasons and proofs.
Euclidean geometry is one of the most foundational parts of mathematics that fit into
a deductive and logical system.
In high school, Euclidean geometry is limited to plane (two dimensional) geometry,
but in University, it gets merged with other disciplines including differential calculus,
topology and solid geometry of three dimensions.
Geometry is used in art, architecture and engineering (along with analytical
geometry) and is used to design structures. It is also used in geography when
drawing maps of the Earth (which requires elliptic geometry).
Anyway, enough with the trivia. Let’s get into it.
1. Circle Geometry
There are 9 theorems taught in grade 11 that you need to know how to prove and
how they work. We will be discussing each theorem below.
But before we do, let’s define a few basic circle terminology:
- Radius: This is the line drawn from the centre of a circle to any point on the
circumference of the circle. (plural: radii)
- Diameter: This is a line passing through the centre of the circle which is twice
the length of the radius.
- Chord: This is a line connecting two points on the circumference of a circle,
i.e a line segment with endpoints on the circumference.
- Tangent: This is the line that touches/intersects the circle at only one point.
- Secant: This is a line passing through two points on a circle.
- Arc: This is a part of a circumference of a circle.
- Segment: The area between the arc of a circle and its chord.
- Sector: This is the area between two radii and an arc of a circle.
- Disk: The area of the whole circle
, 1.1 THEOREM 1(A)
Theorem 1A states that a line drawn from the centre of a circle to the midpoint of a
chord is perpendicular to the chord.
In essence, according to the sketch, if AM = MB then OM ⟂ AB which means ∠M 1 =
∠M2 = 90°
Given: Circle with centre O. M is drawn on chord AB such that it bisects AB, i.e. AM
= MB.
RTP: That OM is perpendicular to AB.
Proof:
Draw radii OA and OB to construct triangles OMA and OMB
In triangles OMA and OMB:
