Draw Transformed Figures

Lesson 5 of 5: Congruence Cluster A

In this lesson:

• Draw images under translations, reflections, and rotations
• Specify sequences of transformations mapping one figure to another
• Verify transformations using the CO.A.4 formal definitions

Learning Objectives for This Lesson

By the end of this lesson, you should be able to:

1. Draw the image of a figure under a given translation using graph paper or ruler
2. Draw the image of a figure under a given reflection using coordinate rules or perpendicular-bisector construction
3. Draw the image of a figure under a given rotation using coordinate rules or compass and protractor
4. Specify a sequence of transformations that maps one figure onto another
5. Verify that a proposed transformation or sequence correctly maps pre-image to image

From Formal Definitions to Actual Drawing

In CO.A.4, you learned what each transformation is:

• Translation: every is parallel to and
• Reflection: is the perpendicular bisector of
• Rotation: on circle at ,

Today: those definitions become construction methods

Transform Every Vertex Independently and Connect

Rule: Transform every vertex independently, then connect in order.

1. List all vertices
2. Apply the transformation to each vertex
3. Plot each image vertex
4. Connect in the same edge order

Every vertex must be transformed — no vertex "follows along"

Translation on a Grid: Triangle ABC

Translation vector : add to each -coordinate, subtract from each -coordinate

Ruler Method: No Coordinates Needed

Method: Copy the translation vector from each vertex using a ruler

For each vertex :

1. Place the ruler at
2. Draw a segment parallel to , same length, same direction
3. The endpoint is

This directly implements the CO.A.4 definition: and

Ruler Method: Walking Through the Steps

Example: Translate by

• At : draw arrow from , parallel to , length →
• At : same →
• At : same →
• Connect in order

Verify:

Check-In: Find Two Image Vertices

Given: Quadrilateral with , ,

Translation vector:

Find and . Try before advancing.

Translation Check-In Answer: Both Vertices Verified

✓

✓

Verify: ✓

Also compute and before connecting the full quadrilateral

Transition: From Translations to Reflections

Translation: every point moves by the same vector

Reflection: each point moves by a different amount toward the mirror line

Two methods:

• Coordinate rule (for axis and standard-line reflections)
• Perpendicular-bisector construction (for any line of reflection)

Reflection Method 1: Coordinate Rules

Reflection Over the y-Axis: Step by Step

Rule: negate the -coordinate, keep unchanged

Reflection Method 2: Perpendicular-Bisector Construction

For any line of reflection — including non-standard lines:

For each vertex :

1. Draw the perpendicular from to → find midpoint
2. Mark on the opposite side:

Implements CO.A.4: is the perpendicular bisector of

Reflection Over y = 1: Worked Example

Mirror , vertex :

• is units above ;
• — 3 units below

Rule:

Reflection Over a Diagonal Line: Construction

Reflect over :

1. Perpendicular from has slope ; meets at
2. equidistant past :

Rule: — coordinates swap ✓

Verifying a Reflection: Two Required Conditions

After constructing, verify both CO.A.4 conditions for each vertex:

• Condition 1: — mirror is perpendicular to
• Condition 2: passes through midpoint of

Measure and — they must be equal

Check-In: Reflect Over a Vertical Line

Given: Vertex . Mirror line: .

Find . Verify both CO.A.4 conditions. Try before advancing.

Answer: Reflection Over x = 3

✓

Condition 1: is horizontal — perpendicular to vertical ✓

Condition 2: Midpoint — lies on ✓

Transition: From Reflections to Rotations

Rotation requires two precise measurements at each vertex:

• The distance from the center (must equal )
• The angle at the center (must equal )

Three methods:

• Coordinate rules (standard angles about the origin)
• Translate-rotate-translate (non-origin center)
• Compass + protractor (general angles)

Rotation Method 1: Standard-Angle Rules About Origin

Positive rotation = counterclockwise (CCW) — "math goes counter"

Rotating 180° About the Origin

Verify: and — both hold ✓

Rotation Method 2: Non-Origin Center

When center ≠ origin, use translate-rotate-translate:

1. Subtract center: — moves to origin
2. Apply the standard rotation rule
3. Add center back:

Equivalent: rotate the vector from to , add back to

Worked Example: Rotate 90° About Non-Origin Center

, CCW about . For :

1. Vector :
2. Rotate CCW:
3. Add :

. Repeat for and .

Compass and Protractor for General Angles

For each vertex and general angle :

1. Set compass to length — this fixes the radius
2. Use protractor at : measure from ray
3. Mark at distance along the new ray

Use this method when is not , , or

Rotation with Compass and Protractor: Walkthrough

, CCW about :

1. — set compass to 5
2. Measure CCW at from rightward ray
3. Mark at distance 5 along the ray

. Verify: ✓

Check-In: Rotate About a Non-Origin Point

Given: . Rotate by CCW about .

Use the translate-rotate-translate method. Find . Try before advancing.

Non-Origin Rotation Answer: P′ at (2, 4)

,

1. Vector :
2. Rotate CCW:
3. Add :

✓ · ✓ · ✓

Transition: When One Transformation Isn't Enough

Sometimes two figures are related by a sequence of transformations:

• Step 1: identify the differences — is the figure flipped? Shifted? Turned?
• Step 2: choose transformations that resolve each difference
• Step 3: apply in order — order matters
• Step 4: verify all vertices match the target

Framework: Analyze Then Compose Transformations

To map onto :

1. Handedness: Is flipped? → reflection needed
2. Position: Is it shifted? → translation may be needed
3. Orientation: Is it turned? → rotation may be needed

Check all three — then compose in order

Specifying a Sequence: Worked Example

Pre-image → image relationship:

1. Reflect over the -axis
2. Translate the result by

Verify all three vertices against the target before claiming the sequence works.

Transformation Order Is Not Commutative

Path 1 — reflect then translate :
— back to start!

Path 2 — translate then reflect:

Different results — order is not commutative

Verify a Sequence by Checking All Vertices

Every vertex must map exactly to the target.

1. Apply step 1 to every vertex — record intermediates
2. Apply step 2 to every intermediate — record finals
3. Check each final against the target — all must match

"Close enough" is not correct in geometry

Check-In: Does Order Matter Here?

Sequence A: Reflect over -axis, then translate by

Sequence B: Translate by , then reflect over -axis

Apply both sequences to . Do you get the same result? Try before advancing.

Order Check-In Answer: Same Result Here

A:

B:

Same result — special case: translation is horizontal, reflection is vertical

Most reflection-translation pairs do not commute

Key Takeaways and Misconception Warnings

✓ Translation: copy vector from each vertex
✓ Reflection: ⊥ bisector for any mirror line
✓ Rotation: exact distance and angle

Every vertex — none follow along
CCW = positive; origin rules only at origin
Find intermediate before step 2
Order matters; verify all vertices exactly

What Comes Next: CO.B and Congruence

CO.B.6: Two figures are congruent iff a rigid motion maps one to the other

• Finding a sequence mapping to proves
• Today's drawing skills become tomorrow's proof tools

CO.B.7-8: SAS, ASA, SSS proved as rigid-motion theorems