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Congruence Concepts

Grade 8 · Geometry · Worksheet 2

  1. A triangle with vertices at (6, 4), (9, 1), and (2, 7) is reflected across the x-axis. What are the coordinates of the reflected vertex that was originally at (9, 1)? Answer: ______________
  2. Liam is designing a triangular garden plot. He places markers at points A(2, 3), B(5, 7), and C(8, 3) on a coordinate grid. He wants to create a second plot that is congruent to the first by applying a sequence of rigid motions: first reflecting the triangle across the y-axis, then translating it 4 units down. What are the coordinates of the vertices of the second triangular plot after these transformations? Answer: ______________
  3. (3x + 2)² - (2x - 1)² = ? Answer: ______________
  4. A triangle with vertices at (6, 4), (9, 8), and (5, 10) is reflected across the x-axis. What are the coordinates of the reflected vertex that was originally at (9, 8)? Answer: ______________
  5. Mason is creating a mosaic pattern on a coordinate grid for his art project. He places a triangular tile with vertices at A(2, 7), B(7, 7), and C(2, 12). To create a congruent tile for the opposite side of the mosaic, he reflects this triangle across the y-axis and then translates the resulting triangle 3 units down. What are the coordinates of the final position of vertex C after both transformations? Answer: ______________
  6. A triangle with vertices at (4, 7), (9, 2), and (1, 5) is reflected across the y-axis. What are the coordinates of the reflected vertex that was originally at (9, 2)? Answer: ______________
  7. A rectangular flag is drawn on a coordinate plane with vertices at (1, 2), (7, 2), (7, 5), and (1, 5). The flag is first reflected across the line y = x, then rotated 90° counterclockwise about the origin. What are the coordinates of the final position of the vertex that was originally at (7, 5)? Answer: ______________
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Answer Key & Explanations

Congruence Concepts · Grade 8 · Worksheet 2

  1. A triangle with vertices at (6, 4), (9, 1), and (2, 7) is reflected across the x-axis. What are the coordinates of the reflected vertex that was originally at (9, 1)? Answer: (9, -1) Solution: Identify the original coordinates of the vertex: (9, 1) Apply reflection across the x-axis. Reflection across the x-axis changes the sign of the y-coordinate while keeping the x-coordinate the same.
    Full step-by-step solution

    Step 1: Identify the original coordinates of the vertex: (9, 1) Step 2: Apply reflection across the x-axis. Reflection across the x-axis changes the sign of the y-coordinate while keeping the x-coordinate the same. Step 3: New x-coordinate = 9, new y-coordinate = -1 Step 4: The reflected vertex coordinates are (9, -1) The answer is (9, -1).

  2. Liam is designing a triangular garden plot. He places markers at points A(2, 3), B(5, 7), and C(8, 3) on a coordinate grid. He wants to create a second plot that is congruent to the first by applying a sequence of rigid motions: first reflecting the triangle across the y-axis, then translating it 4 units down. What are the coordinates of the vertices of the second triangular plot after these transformations? Answer: A'(-2, -1), B'(-5, 3), C'(-8, -1) Solution: A(2, 3) B(5, 7) C(8, 3) Reflection across the y-axis changes the sign of the x-coordinate, while the y-coordinate stays the same.
    Full step-by-step solution

    Let's go step-by-step. --- **Step 1: Original coordinates** We have triangle ABC with: A(2, 3) B(5, 7) C(8, 3) --- **Step 2: Reflection across the y-axis** Reflection across the y-axis changes the sign of the x-coordinate, while the y-coordinate stays the same. Rule: (x, y) → (-x, y) A(2, 3) → A_ref(-2, 3) B(5, 7) → B_ref(-5, 7) C(8, 3) → C_ref(-8, 3) After reflection: A_ref(-2, 3), B_ref(-5, 7), C_ref(-8, 3) --- **Step 3: Translation 4 units down** Translation 4 units down means subtract 4 from the y-coordinate of each reflected point. Rule: (x, y) → (x, y - 4) A_ref(-2, 3) → A'(-2, 3 - 4) = (-2, -1) B_ref(-5, 7) → B'(-5, 7 - 4) = (-5, 3) C_ref(-8, 3) → C'(-8, 3 - 4) = (-8, -1) --- **Step 4: Final coordinates** A'(-2, -1) B'(-5, 3) C'(-8, -1) These match the given correct answer.

  3. (3x + 2)² - (2x - 1)² = ? Answer: 5x² + 16x + 3 Solution: Identify a = (3x + 2) and b = (2x - 1) Apply the difference of squares formula: (3x + 2)² - (2x - 1)² = [(3x + 2) + (2x - 1)] × [(3x + 2) - (2x - 1)] Simplify the first bracket: (3x + 2) + (2x - 1) = 3x + 2x + 2 - 1 = 5x + 1 Simplify the second bracket: (3x + 2) - (2x - 1) = 3x - 2x + 2 + 1 = x…
    Full step-by-step solution

    Step 1: Identify a = (3x + 2) and b = (2x - 1) Step 2: Apply the difference of squares formula: (3x + 2)² - (2x - 1)² = [(3x + 2) + (2x - 1)] × [(3x + 2) - (2x - 1)] Step 3: Simplify the first bracket: (3x + 2) + (2x - 1) = 3x + 2x + 2 - 1 = 5x + 1 Step 4: Simplify the second bracket: (3x + 2) - (2x - 1) = 3x - 2x + 2 + 1 = x + 3 Step 5: Multiply the results: (5x + 1)(x + 3) = 5x × x + 5x × 3 + 1 × x + 1 × 3 = 5x² + 15x + x + 3 Step 6: Combine like terms: 5x² + 16x + 3 The answer is 5x² + 16x + 3.

  4. A triangle with vertices at (6, 4), (9, 8), and (5, 10) is reflected across the x-axis. What are the coordinates of the reflected vertex that was originally at (9, 8)? Answer: (9, -8) Solution: Identify the original coordinates of the vertex: (9, 8) Reflection across the x-axis changes the sign of the y-coordinate, but the x-coordinate remains the same.
    Full step-by-step solution

    Step 1: Identify the original coordinates of the vertex: (9, 8) Step 2: Reflection across the x-axis changes the sign of the y-coordinate, but the x-coordinate remains the same. Step 3: Apply the transformation: x-coordinate remains 9, y-coordinate becomes -8 Step 4: The reflected vertex is at (9, -8) The answer is (9, -8).

  5. Mason is creating a mosaic pattern on a coordinate grid for his art project. He places a triangular tile with vertices at A(2, 7), B(7, 7), and C(2, 12). To create a congruent tile for the opposite side of the mosaic, he reflects this triangle across the y-axis and then translates the resulting triangle 3 units down. What are the coordinates of the final position of vertex C after both transformations? Answer: (-2, 9) Solution: Start with original vertex C at (2, 12). Reflect across the y-axis. The x-coordinate changes sign (2 becomes -2), and the y-coordinate stays the same (12).
    Full step-by-step solution

    Step 1: Start with original vertex C at (2, 12). Step 2: Reflect across the y-axis. The x-coordinate changes sign (2 becomes -2), and the y-coordinate stays the same (12). So after reflection, C' is at (-2, 12). Step 3: Translate 3 units down. Subtract 3 from the y-coordinate: 12 - 3 = 9. The x-coordinate stays the same: -2. Step 4: The final coordinates of vertex C are (-2, 9). The answer is (-2, 9).

  6. A triangle with vertices at (4, 7), (9, 2), and (1, 5) is reflected across the y-axis. What are the coordinates of the reflected vertex that was originally at (9, 2)? Answer: (-9, 2) Solution: Identify the original coordinates of the vertex: (9, 2) Reflection across the y-axis changes the sign of the x-coordinate while keeping the y-coordinate unchanged.
    Full step-by-step solution

    Step 1: Identify the original coordinates of the vertex: (9, 2) Step 2: Reflection across the y-axis changes the sign of the x-coordinate while keeping the y-coordinate unchanged. Step 3: Apply the transformation: new x-coordinate = -9, new y-coordinate = 2 Step 4: The reflected vertex is at (-9, 2) The answer is (-9, 2).

  7. A rectangular flag is drawn on a coordinate plane with vertices at (1, 2), (7, 2), (7, 5), and (1, 5). The flag is first reflected across the line y = x, then rotated 90° counterclockwise about the origin. What are the coordinates of the final position of the vertex that was originally at (7, 5)? Answer: (-5, -7) Solution: Start with the original vertex at (7, 5). Apply reflection across the line y = x. This transformation swaps the x and y coordinates.
    Full step-by-step solution

    Step 1: Start with the original vertex at (7, 5). Step 2: Apply reflection across the line y = x. This transformation swaps the x and y coordinates. (7, 5) becomes (5, 7) Step 3: Apply a 90° counterclockwise rotation about the origin. The rule for this rotation is (x, y) becomes (-y, x). (5, 7) becomes (-7, 5) Step 4: Verify the transformation sequence: Original (7, 5) → After reflection (5, 7) → After rotation (-7, 5) The final coordinates are (-7, 5).