102i - Problem Solving with Similarity Algorithms

Lesson 102 — Problem Solving and Consolidation: Algorithms for Shapes

Strand: Space | Descriptor: AC9M8SP04 | Duration: 45 minutes

Equipment: ruler, ideally squared or dotted paper for sketching similar shapes.

Learning Intentions

• To design a complete, tested algorithm that decides whether two shapes are similar.
• To consolidate the full algorithm cycle: design, test, fix, describe.

Success Criteria

1. Design an algorithm using AA, SSS~ or SAS~ that decides similarity and reports the scale factor.
2. Test the algorithm with a deliberate range of cases, including edge cases.
3. Describe the finished algorithm clearly in plain language.
4. Justify the order of the algorithm’s questions.

Warmup

(5 minutes — congruent or similar? whole class, mini whiteboards)

For each pair, decide: congruent, similar but not congruent, or neither.

1. Two triangles with sides cm and cm.
2. Two triangles with sides cm and cm.
3. Two triangles with sides cm and cm.
4. Two triangles with angles and , one with sides double the other.

Answers: 1. Congruent (SSS, identical sizes); 2. Similar — every side of the second is exactly double the first, so ratios are equal (), but sizes differ; 3. Neither — , , but , so ratios are not all equal; 4. Similar — same three angles (AA, in fact AAA) and the scale factor is .

The key distinction to restate: similar shapes have the same shape (equal angles, proportional sides); congruent shapes additionally have the same size (scale factor ). Congruence is the special case of similarity where the scale factor is .

Activities

Activity 1 — Bridging: the Three Similarity Tests (8 min)

Explicit recap, building directly on Lesson 101’s congruency work.

Test	Condition	Congruency equivalent
AA	Two angles of one triangle equal two angles of the other	(no direct equivalent — angles alone never force congruence)
SSS~	All three pairs of corresponding sides are in the same ratio	SSS (ratio )
SAS~	Two pairs of corresponding sides are in the same ratio, and the included angle is equal	SAS (ratio )

Worked trace — SSS~: Triangle 1 has sides cm. Triangle 2 has sides cm. Match longest to longest, shortest to shortest:

All three ratios equal , so the triangles are similar with scale factor .

The trap to name explicitly: ratios must compare corresponding sides — always match sides by size (smallest-to-smallest, largest-to-largest) or by the angles between them, never by position on the page.

Activity 2 — Main Task: Design a Similarity Algorithm (19 min)

Pairs. This is the assessable design task for AC9M8SP04, consolidating the whole algorithms block (Lessons 100–102).

Design brief. A Year 7 class needs an algorithm that takes the side lengths and angles of two triangles and decides:

• Are they similar?
• If yes, by which test (AA, SSS~ or SAS~), and what is the scale factor?
• If not proven similar, say so.

Your algorithm must:

1. Be written in numbered steps with testable yes/no decisions.
2. Check for AA, SSS~ and SAS~ in a sensible order (justify your choice).
3. Correctly reject an SSA-style ambiguous case, echoing Lesson 101.
4. End every path with a definite output — including the scale factor when similar.

Socratic scaffolding (Polya cycle) for pairs who are stuck:

Prompt	Purpose
Understand the problem. What exactly must the algorithm output for every possible input?	A yes/no similarity decision, the test used, and a scale factor — or “not proven similar”.
What data will it be given?	Three side lengths and/or three angles for each triangle.
Devise a plan. Which similarity test needs the least information to check first?	AA — only angles, no ratio arithmetic — so checking it first is efficient.
What must you do differently for SSS~ compared with SSS in Lesson 101?	Compute and compare ratios, not raw equality of sides.
Carry out the plan. Write step 1 as a full decision, both branches.	E.g. “Do two angles of equal two angles of ? Yes → similar (AA); compute scale factor from any matching side pair. No → step 2.”
How do you compute the scale factor once similarity is confirmed?	Divide a side of the larger (or second) triangle by the corresponding side of the first.
Look back. Trace your finished algorithm on the warmup’s four pairs. Does it agree with your earlier answers?	If not, find the mismatch and repair it before moving on.

Reference solution (teacher — one valid design among several):

1. Do two angles of equal two angles of ? Yes → similar (AA). Compute scale factor from any pair of corresponding sides, if given. Stop.
2. No → For all three corresponding side pairs, is the ratio the same value ? Yes → similar (SSS~), scale factor . Stop.
3. No → Do two corresponding side pairs share the same ratio , and is the angle included between those two sides equal? Yes → similar (SAS~), scale factor . Stop.
4. No → not proven similar. Stop.

Marking emphases: AA first (least computation, most efficient — the halving/efficiency principle from earlier algorithm lessons); step 3 must say “included” angle, exactly the Lesson 101 fault to avoid repeating; every stop state names the scale factor or explicitly withholds one.

Activity 3 — Test, Audit, Refine (8 min)

Swap algorithms with another pair. Build a test table and audit.

Test case	Kind	Expected
Sides and	typical	similar, SSS~,
Angles and , no sides given	typical	similar, AA, no scale factor computable
Sides and	edge	not similar
Sides (ratio ), non-included angle equal, vs matching data	edge	not proven similar (SSA-style trap)
One triangle’s data incomplete (only one side given)	invalid	not proven similar

Run all five through your partner’s algorithm. Record any failures, agree on the smallest fix, and re-test.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. Two triangles have sides cm and cm. Are they similar? State the test and the scale factor.
2. Two triangles share two equal angles but no side lengths are given. Can your algorithm compute a scale factor? Explain.
3. Why must the “included angle” condition appear in an SAS~ step, just as it did for SAS in Lesson 101?
4. Reasoning. Explain, in one or two sentences, why AA alone is sufficient to prove similarity, while SS alone (two sides in ratio, no angle) is not.

Answers: 1. Yes — SSS~, since , scale factor ; 2. No — angles alone confirm the shape matches (similarity) but give no information about relative size, so no scale factor can be computed without at least one matching side pair; 3. Without the included-angle condition, the same SSA-style ambiguity from Lesson 101 applies — two sides in the same ratio with a non-included equal angle can still describe two different shapes; 4. Two matching angles fix all three angles (since angles sum to ), which fixes the triangle’s shape completely regardless of size; two sides in ratio alone say nothing about the angle between them, so the third side (and hence the shape) is not determined.

Common Misconceptions

Misconception	How to pre-empt it
Matching sides by position on the page rather than by size or by the angle between them.	Explicit “smallest-to-smallest” rule in Activity 1’s worked trace.
Believing SS (two sides in ratio, no angle condition) proves similarity.	Exit Q4; contrast directly with AA.
Forgetting the “included” qualifier in SAS~, repeating Lesson 101’s fault.	Marking emphasis in Activity 2; the SSA-style edge case in the test table.
Treating congruence and similarity as unrelated ideas.	Warmup explicitly frames congruence as similarity with scale factor .
Reporting “similar” without a scale factor when one is computable.	Design brief requirement 4: every similar output must attempt a scale factor.
Assuming AA needs a third angle checked.	The angle sum property (Lesson 46/47 knowledge) makes the third angle automatic — worth restating if it comes up.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Two similar triangles have scale factor . If the smaller has area , what is the area of the larger?

Answer

Area scales with the square of the scale factor: .

E2 (AMC Junior style). A triangle with sides is similar to a second triangle with perimeter . Find the sides of the second triangle.

Answer

Perimeter of the first is . Scale factor . Second triangle’s sides: , , .

E3 (Challenge). Explain why “AAA” is never listed as a congruence test, but “AA” is a valid similarity test.

Answer

Matching all three angles fixes a triangle’s shape but says nothing about its size — infinitely many similar triangles share the same three angles. Congruence requires size to match too, so angle information alone can never be enough.

E4 (Challenge). Design (in plain language) one additional decision your Activity 2 algorithm would need to also handle quadrilaterals, not just triangles.

Answer

E.g. “Are all corresponding angles equal and all corresponding sides in the same ratio?” — because unlike triangles, equal angles alone (AA-style) do not guarantee similarity for quadrilaterals (a square and a non-square rectangle share four right angles but are not similar), so both conditions must be checked together.

Homework

1. Two triangles have sides cm and cm. Are they similar? State the test and scale factor.
2. Two triangles share angles and . Are they similar? Can you find a scale factor? Explain.
3. Trace your Activity 2 algorithm on the pair: sides cm with included angle , versus sides cm with included angle . What does it output, and why?
4. Write, in plain language, the smallest fix needed if a classmate’s algorithm checked “two sides in ratio and any equal angle” for SAS~.
5. A triangle with sides is similar to a second triangle. If the second triangle’s shortest side is , find its other two sides.
6. Convert your finished Activity 2 algorithm into pseudocode, using IF … THEN … ELSE IF … ELSE exactly as in Lesson 101.
7. Reasoning. Explain why every congruency test from Lesson 101 has a similarity counterpart, but similarity has the extra AA test with no congruency equivalent.
8. Challenge. Two similar rectangles have areas and . Find the scale factor between them (careful: use area, not a side length).

Answers: Q1 — Yes, SSS~: , scale factor . Q2 — Similar by AA (third angles are both ); no scale factor computable without a matching side pair. Q3 — Not proven similar: the ratio but , so the sides are not in the same ratio — SSS~/SAS~ both fail, and AA does not apply since no second angle is given. Q4 — Replace “any equal angle” with “the angle included between the two ratio-matched sides”. Q5 — scale factor , so the other sides are and . Q7 — congruency is similarity with scale factor exactly , so every similarity test (AA excepted) has a “ratio ” congruency version; AA has no congruency counterpart because angles alone never fix size, only shape. Q8 — area ratio ; scale factor is the square root, .