Download A Guide to Advanced Euclidean Geometry

A Guide to Advanced Euclidean Geometry
Teaching Approach
In Advanced Euclidean Geometry we look at similarity and proportion , the midpoint theorem and
the application of the Pythagoras theorem. The videos included in this series do not have to be
watched in any particular order. Summaries of skills and contexts of each video have been included
in this document, allowing you to find something appropriate, quickly and easily. Each video is short
enough to fit into a lesson, with time left over to discuss the content and some related work.
When teaching this section, it is important to do some integration with aspects that have been
covered before, like ratios and fractions, so that the work would make mathematical sense to the
learners. This is a practical topic and learners should practice as much as possible. Knowledge
should be built conceptually to ensure that they understand what they’re doing. The skills of the
learners will improve if they practice in different contexts.
Video Summaries
Some videos have a ‘PAUSE’ moment, at which point the teacher or learner can choose to
pause the video and try to answer the question posed or calculate the answer to the problem
under discussion. Once the video starts again, the answer to the question or the right
answer to the calculation is given.
Mindset suggests a number of ways to use the video lessons. These include:
 Watch or show a lesson as an introduction to a lesson
 Watch of show a lesson after a lesson, as a summary or as a way of adding in some
interesting real-life applications or practical aspects
 Design a worksheet or set of questions about one video lesson. Then ask learners to
watch a video related to the lesson and to complete the worksheet or questions, either in
groups or individually
 Worksheets and questions based on video lessons can be used as short assessments or
exercises
 Ask learners to watch a particular video lesson for homework (in the school library or on
the website, depending on how the material is available) as preparation for the next days
lesson; if desired, learners can be given specific questions to answer in preparation for
the next day’s lesson
1. Discovering Similarity and Proportion
In this video, we discuss similarity and proportion. We look at conditions that have to be
satisfied for polygons to be similar. We also calculate parts of line segments by using
proportionality.
2. Proportion Theorem
We prove the proportionality theorems that a line drawn parallel to one side of a triangle
divides the other two sides proportionally, including the midpoint theorem. We look at
equiangular triangles and why we say they are equal.
3. Similarity Theorem
In this video we use established results to prove similarity theorem in similar triangles.
4. Similarity Pythagoras Theorem
In this video we use the proven similarity theorem to prove the Pythagoras theorem in
right angled triangles.
Resource Material
Resource materials are a list of links available to teachers and learners to enhance their experience of
the subject matter. They are not necessarily CAPS aligned and need to be used with discretion.
1. Discovering Similarity and
Proportion
2. Proportion Theorem
3. Similarity Theorem
4. Similarity and Pythagoras
Theorem
http://www.mathx.net/proportionsand-similarityhttp://www.onemathematicalcat.org/
Math/Geometry_obj/similarity.htmhttp://www.education.com/studyhelp/article/ratio-proportionsimilarity/http://math.tutorvista.com/geometry/
proportionality-theorem.htmlhttp://math.tutorvista.com/geometry/
proportionality-theorem.htmlhttp://www.pinkmonkey.com/studyg
uides/subjects/geometry/chap5/g05
05401.asphttp://everythingmaths.co.za/grade12/08-euclidean-geometry/08euclidean-geometry-05.cnxmlplus
http://www.khanacademy.org/math/
geometry/right_triangles_topic/pyth
agorean_proofs/v/pythagoreantheorem-proof-using-similarity
http://www.mathwarehouse.com/ge
ometry/similar/triangles/similartriangle-theorems.phphttp://mathandmultimedia.com/2012
/05/21/using-similarity-provepythagorean-theorem/
Worksheets on similarity and
proportion.
Definition for similarity and
examples on similarity
A lesson summary on similarity
and proportion
Triangle proportionality theorem
and its converse
Proportionality practice work
Proportionality study guide
A textbook chapter on similarity.
Pythagorean proof using similarity
Similar triangle theorems
Pythagoras theorem and similarity
proofs.
Task
Question 1
In the diagram, O is the centre of circle. P, Q and R
are points on the circumference of the circle.
Prove that
̂
̂
Question 2
In the diagram, O is the centre of the
circle. A, B, C and D are points on the
circumference of the circle. EOB is a
straight line such that E lies on AD.
AB = BC and ̂
2.1
Calculate, with reasons, ̂ in
terms of x
2.2
Show that EB bisects ̂
2.3
Prove that EOCD is a cyclic
quadrilateral
Question 3
In the figure, ∆ABC has D and E on BC. BD
= 12 cm and DC = 20 cm.
AT : TC = 3 : 1 and AD || TE.
3.1 Write down the numerical value of
3.2 Show that BD is
of BE.
3.3 If FD = 2cm, calculate the length of TE
Question 4
4.1 If BT = , calculate TQ in terms of
4.2 Calculate the numerical value of
4.3 Calculate the numerical value of
Question 5
is a right angled triangle with ̂
D is the
point on AC such that
and E is a point on AB
such that
E and D are joined.
and
5.1
5.2
5.3
Prove that
Calculate BD (leave your answer in surd form)
Calculate AE (leave your answer in surd form)
Question 6
In the diagram M is the centre of the circle.
FEC is a tangent to the circle at E. D is the
midpoint of AB.
6.1
Prove that MDCE is a cyclic quadrilateral
6.2
Prove that
6.3
Calculate CE if AB = 60 cm, ME = 40 cm
and BC = 20 cm
Question 7
In the diagram, AC is the diameter of the circle with
centre O. AC and chord BD intersect at E. AB, BC and
AD are also chords of the circle. OD is joined. AE
If ̂
calculate, with reasons, the size of:
̂
7.1
7.2 ̂
7.3 Show that AE bisects ̂
Question 8
ED is a diameter of a circle, with centre O. ED
is extended to C. CA is a tangent to the circle
at B. AO intersects BE at F. BD//AO. ⏞
8.1 Write down, with reasons, THREE other
angles equal to x
8.2 Determine, with reasons, ̂ in terms
of x
8.3 Prove that F is the midpoint of BE
8.4 Prove that
8.5 Prove that
Task Answers
Question 1
Construction:
̂
̂
̂
ROE
Exterior angle of a triangle
Radii
base angles of an isosceles triangle
OR = OQ
̂ = ̂
̂
̂
̂
̂
̂
Similarly: ̂
̂
̂
Question 2
̂
2.1 ̂
̂
̂
̂
̂
angles in a triangle
̂
̂
=
=
̂
̂
̂
angle at the centre is twice the angle on the
circumference
̂
̂
̂
̂
̂
2.2 .
̂
̂
̂
̂
̂
̂
̂
sum of angles in a triangle
̂
̂
̂
̂
2.3 ̂
̂
̂=
̂
̂=
̂
̂
̂
̂
̂
̂
̂
opposite angles in a cyclic quadrilateral
Question 3
3.1 AT: TC = 3:1
line div. proportional
3.2 Show that BD is
of BE.
DE =15 cm
BD = 12cm
And BE = BD + DE
27 cm
3.3
line div. proportional
TE = 4,5 cm
Question 4
4.1 In
4.2 In
given
4.3
=
=
=
=
=
=
Question 5
5.1
̂
iven
̂
̂
̂
=
̂
+
̂
=
̂
̂
=
̂
angles in a triangle
Given
In
̂
̂
̂
̂
̂
both
3 angle
rd
5.2
= DC.15
But
=
10
= 150
DB =√
= √
5.3 In
Pythagoras
√
= 375
√
In
ED is parallel to BC
√
√
√
√
Question 6
̂
̂
6.1
radius is perpendicular to the tangent
Line from the centre bisects the chord
̂
̂
opposite angles are supplementary
Pythagoras
6.2
6.3
And
Question 7
7.1 ̂
̂
̂
̂
(int. <s
̂
̂
̂
̂
(Angles in the same segment)
̂
̂
7.2
̂
(< at the centre = 2 < at the circumference)
̂
̂
=
̂
̂
(ext. <
̂
̂
̂
=
7.3 In
AE= AE (common)
DE = BE line bisects
̂
̂
(
(s, <, s)
̂
̂
Question 8
8.1 ̂
̂
( base angles of an isosc
̂
̂
̂
(angle in the alternate segment)
( corresponding angles
̂
̂
̂
̂
8.2
̂
8.3
̂
̂
̂
̂
̂
̂
̂
(Angle in a semi circle)
̂
(BD AO)
(angles in a semi circle)
bisects BE (line from centre
is the midpoint of BE
8.4 In
̂
̂
̂
(common)
̂ (= x in 8.1)
̂
̂ (3rd angle)
(<; <; <)
8.5
(8.3)
)
Acknowledgements
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