Common Core Geometry Unit 1 Lesson 5 Homework Answers

Mastering Common Core Geometry Unit 1: Lesson 5 on Congruence and Rigid Transformations

Navigating the foundational concepts of high school geometry can feel like learning a new language, especially when introduced through the structured lens of the Common Core standards. Unit 1, focusing on Congruence, Proof, and Constructions, sets the critical stage for all future geometric reasoning. Lesson 5, typically centered on defining congruence through rigid transformations (translations, reflections, and rotations), is a pivotal moment where abstract definitions meet concrete application. This article provides a comprehensive, concept-driven guide to the types of problems you will encounter in your Common Core Geometry Unit 1 Lesson 5 homework, moving beyond mere answers to build a durable understanding of why those answers are correct.

The Core Concept: Congruence via Rigid Motions

The heart of Lesson 5 is the precise, Common Core-aligned definition: two figures are congruent if there is a sequence of rigid motions (translations, reflections, and rotations) that maps one figure exactly onto the other. A rigid motion is a transformation that preserves distance and angle measure—it does not change the size or shape of the figure, only its position or orientation in the plane. This is not just a definition; it is the fundamental tool for proving congruence.

Your homework will test your ability to:

1. Identify which rigid motion (or sequence) maps one figure onto another.
2. Perform a given rigid motion on a figure, often using coordinates.
3. Describe the transformation using mathematically precise language.
4. Apply the concept to prove triangles or other polygons are congruent.

Decoding Common Homework Problem Types

Type 1: Identifying the Transformation from a Diagram

You will be given two congruent figures, often triangles or quadrilaterals, on a coordinate grid. Your task is to name the single transformation or the shortest sequence that maps the pre-image (original figure) to the image (transformed figure).

Example Problem: Triangle ABC has vertices A(1, 2), B(4, 2), C(3, 5). Triangle A'B'C' has vertices A'(1, -2), B'(4, -2), C'(3, -5). What transformation maps ABC onto A'B'C'?

Step-by-Step Reasoning:

1. Compare Coordinates: Look at the x-coordinates: A(1) -> A'(1), B(4) -> B'(4), C(3) -> C'(3). The x-values are identical.
2. Analyze the Change: The y-coordinates have changed: A(2) -> A'(-2), B(2) -> B'(-2), C(5) -> C'(-5). Each y-value is the opposite of the original.
3. Recall Definitions: A transformation that changes the sign of the y-coordinate while keeping the x-coordinate the same is a reflection across the x-axis.
4. Conclusion: The transformation is a reflection over the x-axis. The rule is (x, y) → (x, -y).

Key Tip: Always check coordinates systematically. A translation adds/subtracts the same value to all x's and all y's. A reflection changes the sign of one coordinate based on the axis of reflection. A rotation around the origin has specific rules: 90° clockwise: (x, y) → (y, -x); 90° counterclockwise: (x, y) → (-y, x); 180°: (x, y) → (-x, -y).

Type 2: Performing a Given Transformation

You are given a figure and a specific transformation rule. You must apply it to find the coordinates of the image.

Example Problem: Quadrilateral DEFG has D(-3, 1), E(0, 3), F(2, 1), G(-1, -1). Perform a translation according to the rule (x, y) → (x + 4, y - 2) and list the coordinates of D'E'F'G'.

Step-by-Step Execution: Apply the rule to each vertex:

• D' = (-3 + 4, 1 - 2) = (1, -1)
• E' = (0 + 4, 3 - 2) = (4, 1)
• F' = (2 + 4, 1 - 2) = (6, -1)
• G' = (-1 + 4, -1 - 2) = (3, -3)

Key Tip: The translation rule (x, y) → (x + a, y + b) means "shift a units horizontally (right if a>0, left if a<0) and b units vertically (up if b>0, down if b<0)."

Type 3: Describing a Sequence of Transformations

This is a higher-order skill. You must find the shortest sequence (often two steps) to map one figure onto another when a single rigid motion won't suffice.

Example Problem: Describe a sequence of transformations that maps triangle PQR onto triangle XYZ, given their coordinates.

Strategic Approach:

1. Orientation & Position: First, look at the overall orientation. Is the image rotated or flipped compared to the pre-image? This often indicates a reflection or rotation is needed first.
2. "Anchor" a Point: Try to map one specific, easy-to-identify vertex first. Can you get P to X with a simple move?
3. Test a Hypothesis: Suppose you reflect first. Apply that reflection mentally or on scratch paper to all points. Does the result look like it's just a translation away from the target? If yes, you have your sequence.
4. Order Matters: The sequence Reflection then Translation is different from Translation then Reflection. You must verify the final result matches the target image exactly.

Common Sequences: A reflection followed by a translation is extremely common. A rotation followed by