MATHS2009: Assignment Support Space Shape and Design

Instructions

All questions must be completed using GeoGebra, and your work must be submitted as .ggb files. Ensure that:

• All constructions are clearly labelled.

• You include extensive comments and explanations using textboxes (use the Text Tool in GeoGebra).

• When finished, download your file via:
  Menu → Download As → GeoGebra File (.ggb)

• Question 1 and Question 2 must be saved as separate files using the naming format:
  “MyName-Question-number.ggb”

Submit both files via the dedicated Moodle dropbox under the Assessment section.

Orthocenter and Related Constructions 

Draw triangle ABC. From each vertex, construct a perpendicular line to the opposite side these are the altitudes.
Find the intersection point of two altitudes using the Intersection Tool and relabel this point as O (the orthocenter).
Measure the angles at vertices A, B, and C.

Answer the following:

• Can the orthocenter lie outside the triangle?
  Explain when this occurs.

• Can the orthocenter lie on the triangle?
  If so, identify the exact situations and the locations.

• Can the orthocenter lie inside the triangle?
  Explain when this occurs.

In triangle ABC, locate and label the intersection points of the altitudes with the sides as:

• D on BC

• E on AC

• F on AB

Construct triangle DEF.

Measure the two angles formed at each vertex of triangle DEF, noting how the altitudes from triangle ABC split these angles.

Answer:

• What do you observe about these paired angles?

Find the circumcenter of triangle DEF by constructing the perpendicular bisectors of its sides. Label the circumcenter as G.

Draw segment GF, then construct a circle centered at G with radius GF using the Circle with Center tool.

This circle will intersect each side of triangle ABC at two points:

• One of the intersection points will be one of D, E, F.

• Identify the other intersection points and label them:

  • H on AB

  • I on AC

  • J on BC

Measure the distances:

• AI

• AH

• BH

• BJ

• CI

• CJ

Answer the following:

• What do you notice about these distances?

• Under what conditions do the orthocenter of triangle ABC and the circumcenter of triangle DEF coincide?

Tessellation with Convex Non-Rectangular Quadrilaterals

• Use one type of quadrilateral.

• Display at least three rows.

• Create the tessellation using transformations such as:

  • Reflections

  • Rotations

  • Translations by vector

• Do not simply copy and paste shapes.

Tessellation with Convex Pentagons

• Use one type of pentagon.

• Display at least three rows.

• Apply transformation tools to demonstrate proper tessellation.

Brief Summary of Assessment Requirements

• Question 1 (Orthocenter and Related Constructions) — construct triangle ABC, its three altitudes and orthocenter O; label and measure vertex angles; locate foot points D, E, F (altitude–side intersections) and build triangle DEF; find circumcenter G of DEF, draw circle with centre G and radius GF, identify secondary intersection points H (on AB), I (on AC), J (on BC) and measure distances AI, AH, BH, BJ, CI, CJ; answer conceptual questions about where the orthocenter can lie (inside/on/outside) and when O and the circumcenter of DEF coincide. Include clear labels and textboxes explaining steps and conclusions.

• Question 2 (Tessellations) — produce two plane tessellations in GeoGebra: one using a single convex non-rectangular quadrilateral and one using a single convex pentagon; show at least three rows for each; generate the tilings with transformations (reflection, rotation, translation by vector etc.) — do not duplicate shapes by copy/paste.

• Technical and submission requirements: use the Text tool to provide explanatory comments; download via Menu → Download As → GeoGebra File (.ggb); name files correctly and submit on time.

How the Academic Mentor Guided the Student

1. Clarify deliverables & file protocol

   • Confirmed file-name convention, separate files per question, and Moodle dropbox procedure. Emphasised use of the Text tool for inline explanations so assessors can follow the constructions.

2. Plan the GeoGebra workflow for Question 1

   • Broke Q1 into micro-steps: (a) draw triangle ABC and construct altitudes, (b) locate O and measure angles, (c) mark D, E, F and build triangle DEF, (d) construct perpendicular bisectors → circumcenter G, (e) draw circle with center G and radius GF, (f) identify H, I, J and measure distances. Agreed the student would annotate each step with short textboxes stating the tool used and the geometric fact being demonstrated.

3. Live demo of GeoGebra tools

   • Demonstrated the exact GeoGebra tools: Perpendicular Line, Intersect Two Objects, Perpendicular Bisector, Circle with Centre, Distance or Length, and Text. Showed how to lock points and use layers to keep the figure tidy. Emphasised accurate use of the measurement tools (angle and distance) and preserving dynamic construction so measurements update if the triangle is moved.

4. Concept coaching for Q1 theory

   • Reviewed key geometry facts to link to measurements: orthocenter location depends on triangle type (acute/obtuse/right), the feet D/E/F form the pedal (altitude) triangle, circumcenter construction for DEF, and the special case where multiple triangle centers coincide (notably the equilateral case). Coach modelled how to phrase conclusions succinctly in textboxes (e.g., “Orthocenter lies inside ⇨ triangle is acute. Orthocenter lies on a vertex ⇨ triangle is right-angled at that vertex. Orthocenter lies outside ⇨ triangle is obtuse.”).

5. Design & construct Q2 tessellations

   • Advised on selecting a single quadrilateral and a single pentagon shape suitable for tessellation. Demonstrated producing a repeat pattern by applying rigid transformations (translate-by-vector, rotation about a point, reflection across a line), and explained why these are preferred over copy/paste (they preserve geometric rules and show understanding of isometries). Asked the student to show at least three rows and to annotate which transformation produced each repetition.

6. Quality-control & mathematical justification

   • Reviewed expected observations: paired angle relationships in DEF, symmetry of intersection distances, and theoretical conditions (e.g., orthocenter = circumcenter in equilateral triangles). Coached the student to justify every observed relation with a one- or two-line theoretical statement in a textbox (linking measured numbers to the geometric theorem).

7. Final checking, labeling and comments

   • Ran through a checklist: all points named, every measurement labelled, explanatory textboxes present, constructions are dynamic (no static screenshots only), file names correct, and download procedure rehearsed.

8. Submission rehearsal

   • Walked the student through, checked filenames, and practised uploading to the Moodle dropbox to avoid last-minute technical issues.

Outcome Achieved and Learning Objectives Covered

Outcome:

• Student produced two complete .ggb files containing dynamic, clearly labelled constructions and explanatory textboxes. Q1 demonstrates the altitudes, orthocenter O, triangle DEF, circumcenter G, circle intersections H, I, J, measured distances and concise theoretical conclusions. Q2 shows two fully formed tessellations (quadrilateral and pentagon) created via geometric transformations rather than copy/paste. Both files were downloaded as and uploaded to Moodle before the deadline.

Learning objectives demonstrated:

• GeoGebra proficiency: correct use of perpendiculars, intersections, perpendicular bisectors, circles, angle/distance tools, transformation tools (translation, rotation, reflection), and effective use of textboxes for exposition.

• Geometric reasoning: understanding and demonstration of properties of altitudes and orthocenter placement (inside/on/outside ↔ acute/right/obtuse triangles), construction and properties of the pedal triangle (DEF), perpendicular bisectors and circumcenter, and the geometric interpretation of intersecting circle points on triangle sides.

• Analytical linking of measurement to theory: measurable evidence (angles, distances) connected to known theorems and special cases (e.g., coincidence of centres in equilateral triangles).

• Understanding of tessellation principles: use of isometries to tile the plane, demonstrating the role of symmetry and transformation groups in creating periodic tilings.

• Academic presentation and assessment readiness: clear labelling, stepwise text explanations, reproducible dynamic figures, correct file export and Moodle submission practices.

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