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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 Mindset Learn Executive Head Content Manager Classroom Resources Content Coordinator Classroom Resources Content Administrator Content Developer Content Reviewer Dylan Busa Jenny Lamont Helen Robertson Agness Munthali Sihlobosenkosi Mpofu Lidwinia Nieuwoudt Produced for Mindset Learn by Traffic Facilities Coordinator Facilities Manager Director Editor Presenter Studio Crew Graphics Cezanne Scheepers Belinda Renney Alriette Gibbs Belinda Renney JT Medupe Abram Tjale Wayne Sanderson This resource is licensed under a Attribution-Share Alike 2.5 South Africa licence. When using this resource please attribute Mindset as indicated athttp://www.mindset.co.za/creativecommons