060i - Congruence, Similarity and Transformations

Lesson 60 — Congruence and Similarity Arising from Transformations

Strand: Space | Descriptor: AC9M8SP01 | Duration: 45 minutes

Learning Intentions

• To identify which transformations (translation, reflection, rotation, dilation) produce congruent images, and which produce similar-but-not-congruent images.
• To connect the congruence and similarity conditions from Lessons 57–59 to the transformations that create them.

Success Criteria

I can:

1. State that translations, reflections and rotations (isometries) always produce images congruent to the original.
2. State that dilations produce images similar to the original, and congruent only when the scale factor is .
3. Identify the transformation (or combination of transformations) that maps one given congruent or similar shape onto another.
4. Explain, using transformations, why the SSS/SAS/AAS/RHS congruence tests and the AA/SAS~/SSS~ similarity tests work.

Warmup

(5 minutes — quick matching, mini whiteboards)

For each before-and-after shape pair, name the transformation that occurred (translation, reflection, rotation, or dilation), then answer: did the size change? did the shape change?

1. A triangle slides units right and units up, with no turning or flipping.
2. A triangle flips across a vertical line, becoming its mirror image.
3. A triangle turns about a fixed point, without changing size.
4. A triangle is enlarged so that every side becomes three times as long.

Discussion: “In Q1–Q3, the shape’s size and angles never changed — only its position or orientation did. In Q4, both the size and the shape’s proportions changed together (or did they?). Today we pin down exactly which transformations preserve congruence, and which only preserve similarity.”

Activities

Activity 1 — Explicit Instruction: Isometries Preserve Congruence (12 min)

I do: Define isometry (“equal measure”) — a transformation that preserves distances between every pair of points. Translation, reflection and rotation are all isometries. Since every side length is preserved, and angles are built from side lengths meeting at a point, every angle is preserved too — so the image is congruent to the original, by SSS (all three sides automatically match).

Coordinate example: has , , . Translate by the vector : every point moves the same amount, so , , .

Every side length is preserved under translation, so by SSS. This is exactly why SSS guarantees congruence: if all three sides match, there is always some isometry mapping one triangle exactly onto the other.

We do: Together, reflect with , , across the -axis, giving , , . Check that and , confirming congruence.

You do: For each pair of congruent shapes, name the transformation and describe it fully (a translation vector; a mirror line; or a centre and angle of rotation).

1. with , , maps to with , , — a quarter turn about the origin.
2. with , , maps to with , , .

Activity 2 — Explicit Instruction: Dilation and Similarity (12 min)

I do: Define dilation (enlargement): a transformation with a centre point and a scale factor , where every point’s distance from the centre is multiplied by . Unlike isometries, a dilation multiplies every length by — angles stay the same (so the shape is preserved), but size changes (unless ). This means the image is similar to the original, congruent only when .

Coordinate example: dilate with , , from the origin with scale factor : multiply every coordinate by , giving , , .

Every side is exactly doubled, so the ratio of corresponding sides is constant () — this is SSS~, confirming . This is why AA and SSS~ guarantee similarity: a dilation preserves angles exactly while scaling every side by the same factor.

We do: Together, dilate with , , from the origin with scale factor . Find the image coordinates and verify the side ratio equals .

You do:

1. Dilate with , , from the origin with scale factor . Find the image coordinates and the ratio of corresponding sides.
2. A shape and its image have corresponding sides cm and cm. State the scale factor, and whether the shapes are congruent or similar (not congruent).
3. A shape and its image have corresponding sides cm and cm, but the image required a reflection to align with the original. Are the shapes congruent? Explain.

Activity 3 — Inquiry Task: Name the Transformation (11 min)

Pairs, then whole-class share.

Triangle has , , . Triangle has , , . Determine whether and are congruent or similar, then describe the single transformation that maps one exactly onto the other.

Socratic scaffolding:

Prompt	Purpose
Understand: what must you check first, before naming a transformation?	Whether the shapes are the same size (congruent) or only the same shape (similar) — this determines whether an isometry or a dilation is needed.
Calculate the side lengths of both triangles.	, , ; , , .
Compare the ratios of corresponding sides.	, , — constant ratio, so similar, not congruent (SSS~).
Devise a plan — what transformation produces a similar (not congruent) image?	A dilation, since only dilations change size while preserving shape.
Carry it out — find the centre and scale factor.	Both triangles share vertex , and every other point is exactly twice as far from it in — centre , scale factor .
Looking back — how would this change if ‘s vertices were instead , , ?	The triangles would still be similar (same side ratios), but the transformation would no longer be a simple dilation from the origin — this previews the idea that some mappings combine multiple transformations, which Lesson 61 explores formally.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. Name one transformation that always produces a congruent image.
2. Name the transformation that can produce an image that is similar but not congruent, and state the condition on its scale factor for the image to actually be congruent.
3. A shape is rotated about a point. Are the original and image congruent or similar-only? Justify.
4. Reasoning. Explain, using the idea of preserved distances, why a dilation with scale factor produces an image that is congruent (not just similar) to the original.

Answers: 1. Any of translation, reflection, rotation; 2. Dilation; congruent only when ; 3. Congruent — rotation is an isometry, so all distances (and therefore all side lengths and angles) are preserved; 4. When , every distance from the centre is multiplied by , i.e. left unchanged — so every side length is preserved exactly, making the image identical in size as well as shape, which is the definition of congruence.

Common Misconceptions

Misconception	How to pre-empt it
Any transformation preserves size.	Explicitly separate transformations into two groups: isometries (translation, reflection, rotation — size preserved) and dilations (size changed unless ).
A reflection produces a “different shape,” not a genuine congruent copy.	Measure corresponding sides and angles before and after reflecting to confirm they are identical — only orientation (left/right handedness) changes.
Dilation always enlarges the shape.	Show a dilation with , which shrinks the shape — “dilation” describes any uniform scaling, not only enlargement.
A negative scale factor is not allowed.	Briefly note that produces a rotation of the original (still congruent) — negative scale factors are valid and reverse the direction from the centre.
AAA can prove congruence because “the shapes look the same.”	Connect back to dilation: a dilation preserves all angles but changes every length, giving a concrete reason why equal angles alone (AAA) cannot guarantee equal size.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). A triangle is translated units right and reflected across a horizontal line. Is the final image congruent to the original? Justify your answer using the definition of isometry.

Answer

Yes — both translation and reflection are isometries, and a composition of isometries is still an isometry (distances are preserved at each step), so the final image is congruent to the original.

E2 (Kangaroo style). A shape is dilated with scale factor from a centre point. If a side of the original shape is cm, what is the length of the corresponding side in the image?

Answer

E3 (Challenge). A triangle is dilated by scale factor , then the image is dilated again by scale factor . Is the final shape congruent to the original triangle? Justify using the combined scale factor.

Answer

Combined scale factor , so the final shape has every side length restored to its original value — it is congruent (in fact, identical in position) to the original triangle.

Homework

1. State whether each transformation always produces a congruent image, or only a similar image: (a) rotation (b) dilation with (c) reflection (d) dilation with .
2. A triangle with vertices , , is translated by the vector . Find the coordinates of the image and state whether the image is congruent to the original.
3. A triangle with vertices , , is dilated from the origin with scale factor . Find the image coordinates and the ratio of corresponding side lengths.
4. Explain, using the language of transformations, why a photocopier set to “150%” produces a similar (not congruent) copy of a page.
5. Reasoning. Explain why every congruence test from Lesson 57 (SSS, SAS, AAS, RHS) can be understood as “there exists an isometry mapping one triangle exactly onto the other,” while every similarity test from Lesson 58 (AA, SAS~, SSS~) can be understood as “there exists a dilation (possibly combined with an isometry) mapping one triangle onto the other.”
6. Challenge. Triangle has , , . Triangle has , , . Determine whether the triangles are congruent, and if so, describe the single transformation mapping onto .

Answers: 1. (a) congruent (b) similar only (c) congruent (d) congruent; 2. Image: , , ; congruent, since translation is an isometry; 3. Image: , , ; ratio of corresponding sides ; 4. A photocopier enlargement is a dilation with scale factor — angles and proportions are preserved (same shape), but every length is multiplied by (different size), giving a similar, not congruent, copy; 5. In each congruence test, the matching sides/angles are exactly what’s needed to guarantee a distance-preserving mapping (isometry) exists between the two triangles; in each similarity test, the matching angles or proportional sides guarantee a mapping exists that preserves shape while allowing a uniform size change (a dilation, possibly followed by an isometry to reposition it); 6. Side lengths: ; — matching, so congruent by SSS. The mapping is a clockwise rotation about the origin (since and under a clockwise rotation).