3.4 Transformations and Constructions

Why This Section Matters

Transformations and constructions sit at the intersection of secondary geometry standards and classroom practice. Praxis 5165 tests rigid motions (translations, reflections, rotations), dilations, and whether a described classroom activity matches a true compass-and-straightedge construction. About one-quarter of exam questions are task-of-teaching scenarios — expect items where you correct a student's belief that sliding a figure changes its size.

Understanding which transformations preserve distance helps you answer both coordinate-image items and pedagogy questions about congruence.

Rigid Motions Preserve Congruence

A rigid motion (isometry) preserves distance and angle measure. The image is congruent to the preimage.

Worked Example: Rotation

Rotate (3, −2) 90° counterclockwise about the origin.

Rule: (x, y) → (−y, x). Image: (2, 3).

Check: the point started in Quadrant IV and should land in Quadrant I after a 90° CCW turn — (2, 3) fits.

Worked Example: Reflection Composition

Reflect (4, −1) over the y-axis, then over the x-axis.

After y-axis reflection: (−4, −1). After x-axis reflection: (−4, 1).

Two reflections can equal a rotation in some cases; on Praxis, apply rules step by step unless the stem asks for a single equivalent transformation.

Worked Example: Translation Then Reflection

Translate (1, 4) by ⟨−2, 3⟩, then reflect over the x-axis.

After translation: (1 − 2, 4 + 3) = (−1, 7).

After reflection over x-axis: (−1, −7).

Order matters: translation then reflection generally differs from reflection then translation.

Dilations and Similarity

A dilation with scale factor k > 0 stretches or shrinks about a center. Side lengths multiply by |k|; areas multiply by k².

If k < 0, the figure is also rotated 180° about the center — less common on Praxis but worth noting.

Dilation is the transformation behind similar (non-congruent) figures: same shape, different size.

Compass-and-Straightedge Constructions

Classical constructions allowed with only an unmarked straightedge and compass:

• Copy a segment or angle
• Bisect a segment or angle
• Perpendicular through a point on a line (or off a line)
• Parallel line through a point
• Equilateral triangle inscribed in a circle
• Square inscribed in a circle

Not valid as pure constructions: measuring with a protractor or ruler marked in units, folding (paper), or using a graphing calculator trace.

Teaching Scenario Trap

A student says translating a figure 4 units right and 2 units down changes its area. Best response: a translation preserves lengths and angle measures, so the image is congruent to the original — area is unchanged.

Transformations as Proof Tools

Many congruence proofs can be reframed: showing a reflection maps one triangle onto another establishes congruence without listing six matching parts.

On Praxis, you are more likely to compute an image point or identify the transformation type than to write a full proof.

Symmetry

• Line symmetry: reflection axis maps the figure onto itself.
• Rotational symmetry: rotation about a center maps the figure onto itself (order n for 360/n degrees).

Regular hexagons have six lines of symmetry and rotational symmetry of order 6. Identifying symmetry helps classify figures quickly.

Mapping Notation

Function notation T(x, y) = (x + 3, y − 2) describes a translation. Composition T₂ ∘ T₁ means apply T₁ first, then T₂ — order matches standard function composition.

Common Traps

• Confusing (x, y) → (y, x) with a 90° rotation (that is reflection across y = x).
• Assuming any movement changes area (only dilations with k ≠ 1 change size).
• Calling protractor measurement a construction when the stem specifies classical tools only.
• Applying the wrong rotation direction (CW vs CCW).
• Forgetting that reflections reverse orientation unless followed by another reflection.

Section Checklist

• Classify the transformation before applying a rule — rigid vs dilation.
• For teaching items, tie the correction to an invariant (distance, angle, area).
• For constructions, verify the method uses only compass and straightedge steps.
• Track composition order when multiple transformations are applied.

Dilations on the Coordinate Plane

A dilation with center at the origin and scale factor 3 maps (2, −4) to (6, −12). Distances multiply by 3; perimeter triples; area multiplies by 9. If the center is not the origin, translate so the center moves to the origin, dilate, then translate back — a three-step process rarely required on Praxis unless coordinates are chosen conveniently.

Construction Sequence: Perpendicular Bisector

To construct the perpendicular bisector of segment AB: draw arcs of equal radius centered at A and B that intersect above and below the segment; connect the two intersection points. Every point on the resulting line is equidistant from A and B — the foundation for copying distances and building congruent triangles.

Link to Praxis Practice

Rotation and translation teaching items appear in the geometry bank under transformations. When a stem asks for the best instructional response, anchor your answer to an invariant (distance, angle measure, area) rather than to coordinate sign rules alone.