Identify all lines of symmetry in complete figures
Test and draw every mirror line of a complete given figure, including oblique axes, no-axis examples, concavity and true decoration.
Exam scope and assessment notes
Foundation and Higher share the one published F:4.3A requirement: identify any lines of symmetry and the order of rotational symmetry of a given two-dimensional figure. The Higher table says See Foundation Tier; there are no additional H4.3 rows. Both complete-figure capabilities matter. This chosen three-lesson split requires the lines, rotational-turn and comparison lessons, each with substantive teaching, a worked example and every Guided, Independent and Transfer stage, before the one structural link is retained. Three lessons, twelve original worked/practice cases and twenty original figures are StudyForA’s design, not a syllabus-prescribed count. The official note asks learners to name a quadrilateral with no mirror line and rotational order2; the additional learner completion A(0,0),B(5,0),D(2,3) to find C and the midpoint is chosen original practice within SYM3, not a new source row or a mandatory formal construction. Keep C, the completed outline and its centre/reference result hidden until the learner has attempted the completion. Draw and test all full mirror lines on the entire supplied figure; equal-area division, a diagonal or one successful guess is insufficient. Keep a fixed original and separate rotated tracing, choose/test the centre and record all distinct matches in one turn. Count identity once:0° and360° are the same position. Finite order1 means no nontrivial return before360°, never order0; a full-turn test anchor is not a uniquely determined nontrivial centre. A blank circle has infinitely many mirror axes and matches at every angle about its centre, so it has no finite rotational order or least positive matching angle; never force it into a finite-order integer answer. Preserve true boundary indentations, disks/bolts, category identity and radii; explanatory vertex labels and external legends are not physical decoration unless stated. Rigid page orientation changes preserve counts while moving axes and centres; they are not themselves self-symmetry tests. The scalene seed is (0,0),(5,0),(1,2); the oblique kite is its separately supplied turned fixture. Its numerical reference-grid guidance is reserved for the attempted comparison, rather than required trigonometry. A fixed reference diagram, count choice or acknowledgement does not establish that the learner drew, tested or justified the figure. Written work is not automatically marked. Attempt records and revealed references do not award marks or establish mastery. These are structural links, not exhaustive subskill approval; specialist approval and formal assessment validation remain unestablished. Source assessment notes allow tracing paper and say diagrams need not be to scale; do not measure them unless instructed. Reflection/rotation overlays do not complete5.2 transformations,4.5 instrument/scale/compass constructions,4.11 similarity or the full H4.7 reasoning programme. No blanket formal-proof waiver is imported from unrelated circle-theorem notes. The wider course remains incomplete. Publisher navigation is separate evidence: the only literal F4.3A chapter tag is Book2 Shape and Space6 on map physical10, pages128–160; public uncorrected proof contents start128 and next chapter161 agree. The map supplies no explicit4.3 Hub tag; nearby Hubs58/59,36/37/38 must not be retagged as symmetry. No Book1 chapter has a literal4.3 tag. Limited Book1 printed64 triangle discussion is not a substitute tag: its generic right-triangle zero-lines wording is limited to the drawn scalene case, and its no rotational symmetry wording means finite order1 for non-equilateral examples, with the inclusive equilateral exception preserved. The relevant Book2 mapped chapter body is absent from the public sample; no explicit contents symmetry heading locates every skill. Full final-book and Teaching Hub body/pagination alignment remains unverified; chapter links confer no full-book or modular teaching credit.
Before you begin
- Read polygon boundaries, equal-side marks and right-angle marks.
- Distinguish a complete straight line from a ray or boundary segment.
- Use tracing or folding as an instructed check without treating an unscaled sketch as exact measurement.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
For a non-square 6 cm by 4 cm rectangle, which candidate is a mirror line?
What you will learn
- Use reflection or folding to test the whole boundary rather than merely equal areas.
- Draw all distinct symmetry lines, including oblique lines, and justify failed candidates.
- Distinguish a blank circle’s infinitely many axes from the symmetries of a physically decorated design.
Use three complete outline cards and keep each boundary fixed while testing a reflected tracing. Card R is a 6 cm by 4 cm rectangle, with corners A(−3,−2),B(3,−2),C(3,2),D(−3,2). Card S is a non-square rhombus with diagonals 8 cm and 6 cm, with corners A(−4,0),B(0,−3),C(4,0),D(0,3). Card P is the oblique parallelogram A(0,0),B(6,0),C(8,3),D(2,3). Join each list in boundary order. Draw every mirror axis on each card, naming its location and showing a reflection or fold check. Explain why a diagonal splitting P into equal-area halves does not automatically make a mirror line. Use the coordinates to locate the cards, not to impose a distance-formula method.
Draw and test on paper first. The fixed original stays still while the separate orange tracing moves. All comparison panels share one drawing-unit scale. Include every inward corner and physical disk. A successful numeric trial is evidence about this fixed drawing, not a complete proof or an automatic mark.
Candidate fields always use the original worksheet grid. The entire grid, figure and candidate tracing turn or reflect together in the alternative view. Changing figure, case or view clears your trials and references.
Click a named corner on the active board to use its exact location. Other clicks use the nearest half-unit grid point, with keyboard fields available below. Points never snap to a reference axis or centre. Select a different point role before placing it.
Card R · active drawing
- The supplied card is a non-square rectangle.
- Adjacent side lengths are 6 cm and 4 cm.
- AB: 6 cm
- BC: 4 cm
My mirror-line candidate
Accepted tested lines: 0 of 0 distinct candidates recorded. Reversing or moving points along the same line does not add a second axis. These tests do not establish that you found every axis.
My rotation candidate
Positive turns are anticlockwise in the original worksheet grid. Complete reflected displays reverse their screen direction. Keep records grouped by the centre you supplied; do not combine different centres into one order.
The board retains your last submitted valid numeric trial; invalid field entries do not move that tracing.
Compare other given figures at the same scale
Card S
- The supplied card is a non-square rhombus.
- The given diagonal lengths are AC=8 cm and BD=6 cm.
- AC: 8 cm
- BD: 6 cm
Card P
- The supplied outline is an oblique parallelogram.
A mirror line must preserve the whole figure
A line of symmetry is a full mirror line across which reflection maps the entire figure onto itself. Imagine folding the paper along that line, or reflect a separate tracing while keeping the original fixed. Every boundary segment and every true feature must coincide with a matching part. A line that divides the area equally need not pass this test.
Draw the candidate as one complete straight line, extending through and beyond the figure, rather than counting its two rays separately. Name where it passes, such as opposite side midpoints or two vertices. Try the reflection before accepting it: a failed corner, edge, indentation or coloured mark is enough to reject that candidate. A familiar polygon name can suggest candidates, but the complete given figure decides.
Find every axis, including sloping ones
A non-square rectangle has the two lines through midpoints of opposite sides; its diagonals are not mirror lines because the adjacent side lengths differ. A non-square rhombus instead has its two diagonals as axes. A square combines both types and has four. Do not turn the correct axis for one family into a rule for all quadrilaterals.
A kite with two different-length adjacent equal-side pairs has the diagonal through the vertices where the pairs meet as its one mirror line. In an equilateral triangle, each vertex-to-opposite-midpoint line works, giving three. A non-equilateral isosceles triangle has one, and a scalene triangle none. In particular a right triangle can be isosceles or scalene, so the word right does not determine a single mirror-line count.
Keep indentations and physical features
A concave boundary must be reflected in full. For a T-shaped outline, a candidate line must match both the broad top bar and the narrow stem, including the inward corners. Turning the page does not remove an axis; it turns the correct line with the figure. Horizontal and vertical are descriptions of a displayed orientation, not requirements in the definition.
Treat a printed disk, hole or colour as part of the design when the question says it is physical. Vertex labels and a separate explanatory legend normally identify parts without becoming decoration themselves. A blank circular region has every line through its centre as an axis, so there are infinitely many; draw representative diametral lines and state the whole family. One off-centre printed disk usually removes most of those axes because its centre must also be fixed by reflection.
Sources for this lesson
Pause and explain
A circular badge has exactly one small blue disk away from its centre. Which description gives every mirror line of the full design?
Worked example
Use three complete outline cards and keep each boundary fixed while testing a reflected tracing. Card R is a 6 cm by 4 cm rectangle, with corners A(−3,−2),B(3,−2),C(3,2),D(−3,2). Card S is a non-square rhombus with diagonals 8 cm and 6 cm, with corners A(−4,0),B(0,−3),C(4,0),D(0,3). Card P is the oblique parallelogram A(0,0),B(6,0),C(8,3),D(2,3). Join each list in boundary order. Draw every mirror axis on each card, naming its location and showing a reflection or fold check. Explain why a diagonal splitting P into equal-area halves does not automatically make a mirror line. Use the coordinates to locate the cards, not to impose a distance-formula method.
Show the worked solution
- On R, reflect across the horizontal and vertical lines through the rectangle’s centre. Each swaps equal opposite sides and the matching corners, giving its two midlines. A diagonal reflection would interchange unequal 6 cm and 4 cm directions, so the diagonal candidates fail.
- On S, reflect across AC and BD. The two equal-side halves on either side of each diagonal coincide, so both diagonals are axes. The lines through opposite side midpoints do not preserve this non-square rhombus.
- On P, a reflection across a diagonal or a centre midline leaves some corners or sloping edges unmatched. Its oblique, non-rectangular, non-rhombus outline has no mirror line.
- The diagonal of P does split it into equal-area triangles, but reflection does not superimpose those complete halves. Equal area therefore supplies no replacement for the overlap test. The line counts for R,S,P are 2,2,0.
Answer R has 2 axes, the two opposite-side midlines. S has 2 axes, its diagonals AC and BD. P has 0 mirror axes: equal-area halves do not guarantee reflection overlap. These are conclusions about the complete specified cards.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Use four complete marked outline cards: kite A(0,5),B(−3,0),C(0,−2),D(3,0), whose different-length equal-side pairs meet at A,C; an equilateral triangle with all three equal-side marks; the isosceles right triangle A(−2,0),B(2,0),C(0,2); and the scalene right triangle A(0,0),B(6,0),C(0,4). Join the coordinate lists in boundary order and retain the right/equal-side marks. Draw every full mirror line and test it by reflection of the complete outline. Explain why right triangle alone would not tell you the exact line count.
Give me a hint
Keep the kite’s different top/bottom portions, and separate an isosceles right triangle from a scalene right triangle.
Compare my reasoning
- In the kite, the equal adjacent pairs meet at A,C. Reflect across AC: B and D swap while A,C stay on the line, so this is one axis. Reflecting across BD would move the unequal top and bottom portions onto the wrong parts, so BD is not an axis.
- For the equilateral triangle, draw each of the three vertex-to-opposite-midpoint lines. Each exchanges two equal sides and preserves the complete triangle, so all three work.
- For the isosceles right triangle, draw the line from its right-angle vertex C to the midpoint of hypotenuse AB. Reflection exchanges the equal legs, giving one axis. The scalene right triangle has unequal side lengths and no reflection exchanging its full boundary, giving none.
- Thus the counts in the stated order are 1,3,1,0. A right angle alone does not say whether the two legs are equal; that missing condition changes the answer. Under the inclusive isosceles definition, an equilateral triangle is the three-axis special case, not a universal one-axis example.
Kite: 1 axis, AC. Equilateral triangle: 3 vertex-to-opposite-midpoint axes. Isosceles right triangle: 1 axis from the right vertex to the hypotenuse midpoint. Scalene right triangle: 0. The right-angle condition alone does not select between the last two results.
Look for these in your work
- Draw full candidate lines on all four given figures without treating their rays as separate axes.
- Accept exactly the axes that map the entire marked boundary onto itself.
- Explain the 1,3,1,0 counts and the missing equality condition in a generic right triangle.
Use unannotated complete outline cards. The concave T boundary is A(−3,2),B(3,2),C(3,0),D(1,0),E(1,−3),F(−1,−3),G(−1,0),H(−3,0), closed to A. The scalene triangle is A(0,0),B(5,0),C(1,2), closed to A. Draw and test every mirror axis, challenging vertical and diagonal guesses. Explain which indentation or side correspondence rejects a false candidate; do not replace the entire T by only its top bar. Record one rejected candidate with a specific boundary point that fails to land on its reflected partner.
Give me a hint
Reflect the whole boundary, including every inward corner; a triangle reflection would need a matching equal-side pair.
Compare my reasoning
- Trace all eight sides of the T, including its inward corners. Reflection across the vertical line through the middle of the bar and stem exchanges left and right boundary parts, so that full line is an axis.
- A horizontal reflection would send the upper bar to the lower stem region, and a diagonal reflection would turn the broad bar into an unmatched side feature. These failures show why the complete indentation and stem matter. The T has one axis, not several inherited from a rectangle drawn around its bar. For a specific failed point, the horizontal candidate y=0 sends A(−3,2) to (−3,−2), which is not on the original T boundary.
- For the triangle, test a line through each vertex towards the opposite side. A triangle mirror would fix one vertex and exchange the other two, forcing two corresponding side lengths to agree. This given scalene triangle has three unequal sides, so none of those candidates works.
- Record one axis for the T and none for the triangle. Leave failed candidates visibly distinguished from accepted axes, rather than presenting a guessed diagonal as a second solution.
T: 1 mirror line, through the middle of its bar and stem (vertical in this orientation). Scalene triangle: 0. The T’s whole indented boundary must match, and a scalene triangle cannot exchange two unequal sides under reflection. For example, reflection in y=0 sends A(−3,2) to (−3,−2), which is not on the T boundary, so that candidate fails.
Look for these in your work
- Draw and test the T’s one axis while rejecting misleading bar-only and diagonal matches.
- Justify no axis for the scalene triangle using a side correspondence.
- Keep every indentation and closing side in the tracing rather than simplifying the outline.
Two circular badge designs have the same outer radius of 4 drawing units. One is blank. The other has exactly one small blue disk of radius 0.3 centred 2 units directly above the outer circle’s centre. Keep that blue disk as a true physical part of the second badge, while centre labels are explanatory. Draw representative mirror lines on the blank badge and describe all of them; draw every mirror line on the printed badge. Use a reflected tracing to show why at least one discarded line fails for the complete printed design.
Give me a hint
Test the physical blue disk as well as the outer curve. A line fixing a circular disk must pass through its centre.
Compare my reasoning
- On the blank circle, any full line through its centre reflects the whole circular region onto itself. Draw several diametral examples, including an oblique one, and state that every centre-line works: the line count is infinitely many, not the number of examples drawn.
- For the printed badge, the outer circle still requires an axis through its centre. The single off-centre blue disk must also map onto itself, so the same line must pass through the blue disk’s centre.
- Only one full line passes through both distinct centres. In the supplied orientation it is vertical, and reflecting across it preserves both circular regions.
- A horizontal centre-line sends the blue disk to an unprinted position below the centre, so it fails. The printed badge has exactly one axis; ignoring the decoration would give the wrong infinite count.
Blank badge: infinitely many axes, every line through the circle centre. Printed badge: exactly 1, the line through both circle centres. A horizontal reflection fails because the single blue disk would move to an unprinted location.
Look for these in your work
- Draw representative centre-lines and state the infinite family for the blank badge.
- Draw the printed badge’s unique common-centre line.
- Demonstrate a failed reflection using the actual off-centre blue disk.
Common mistakes
- Accepting a line because it divides the area equally or because it is a diagonal.
- Counting only vertical or horizontal guesses, or ignoring inward boundary details.
- Ignoring actual decoration, treating bookkeeping labels as decoration, or giving a finite count for every centre-line of a blank circle.
What can you explain now?
A diagonal divides a given oblique non-rectangular, non-rhombus parallelogram into two equal-area triangles. Is that enough to call the diagonal a line of symmetry? Describe the required test.
Compare with the explanation
No. Reflect or fold the complete figure across the diagonal and test full coincidence.
The two triangles can have equal area while their reflected boundaries do not coincide. A mirror line requires every corresponding boundary part to match, not merely two equal areas.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Explain a complete-figure reflection test. Contrast rectangle and rhombus axes, give a no-axis example, and explain what an off-centre physical feature does to a blank circle’s axes.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.