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

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

http://www.mathx.net/proportions-

and-similarity-

Worksheets on similarity and

proportion.

http://www.onemathematicalcat.org/

Math/Geometry_obj/similarity.htm-

Definition for similarity and

examples on similarity

http://www.education.com/study-

help/article/ratio-proportion-

similarity/-

A lesson summary on similarity

and proportion

2. Proportion Theorem http://math.tutorvista.com/geometry/

proportionality-theorem.html-

Triangle proportionality theorem

and its converse

http://math.tutorvista.com/geometry/

proportionality-theorem.html-

Proportionality practice work

http://www.pinkmonkey.com/studyg

uides/subjects/geometry/chap5/g05

05401.asp-

Proportionality study guide

3. Similarity Theorem

http://everythingmaths.co.za/grade-

12/08-euclidean-geometry/08-

euclidean-geometry-05.cnxmlplus

A textbook chapter on similarity.

4. Similarity and Pythagoras Theorem

http://www.khanacademy.org/math/

geometry/right_triangles_topic/pyth

agorean_proofs/v/pythagorean-

theorem-proof-using-similarity

Pythagorean proof using similarity

http://www.mathwarehouse.com/ge

ometry/similar/triangles/similar-

triangle-theorems.php-

Similar triangle theorems

http://mathandmultimedia.com/2012

/05/21/using-similarity-prove-

pythagorean-theorem/

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 Prove that

5.2 Calculate BD (leave your answer in surd form)

5.3 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

Construction: ROE

Exterior angle of a triangle

OR = OQ Radii

= base angles of an isosceles triangle

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 1

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

3rd

angle

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

6.2 Pythagoras

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 (Angle in a semi circle)

8.3 (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

Mindset Learn Executive Head Dylan Busa

Content Manager Classroom Resources Jenny Lamont

Content Coordinator Classroom Resources Helen Robertson

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Produced for Mindset Learn by Traffic

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