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Congruence Concepts

Grade 8 · Geometry · Worksheet 2

  1. A triangle with vertices at (6, 8), (10, 12), and (4, 15) is reflected across the x-axis. What are the coordinates of the reflected vertex that was originally at (10, 12)? Answer: ______________
  2. (3x + 2)² - (2x - 1)² = ? Answer: ______________
  3. √(x² - 10x + 25) = 8 Answer: ______________
  4. Liam is designing a triangular garden plot using congruent triangles. He places triangle ABC with vertices at A(2, 3), B(5, 3), and C(2, 7). He wants to create a congruent triangle DEF by applying a sequence of rigid motions: first a translation 4 units right and 2 units down, then a reflection across the x-axis. What are the coordinates of the final triangle's vertices? Answer: ______________
  5. A triangle with vertices at (-6, 8), (9, -5), and (4, 11) is reflected across the x-axis. What are the coordinates of the reflected vertex that was originally at (9, -5)? Answer: ______________
  6. A triangle with vertices at (2, 3), (5, 7), and (8, 3) is reflected across the x-axis. What are the coordinates of the new vertices? Answer: ______________
  7. A rectangular flag is drawn on a coordinate plane with corners at (1, 2), (7, 2), (7, 5), and (1, 5). The flag undergoes a sequence of rigid motions: first it is reflected across the line y = x, then it is rotated 90° counterclockwise about the origin. What are the coordinates of the final position of the corner that was originally at (7, 5)? Answer: ______________
  8. Liam is designing a triangular garden plot. He uses a coordinate grid to plan the layout, with the triangle having vertices at A(2, 1), B(5, 1), and C(3, 4). He wants to create a second identical plot by reflecting the triangle across the y-axis and then translating it 3 units down. What are the coordinates of the vertices of the second triangular garden plot after these two transformations? Answer: ______________
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Answer Key & Explanations

Congruence Concepts · Grade 8 · Worksheet 2

  1. A triangle with vertices at (6, 8), (10, 12), and (4, 15) is reflected across the x-axis. What are the coordinates of the reflected vertex that was originally at (10, 12)? Answer: (10, -12) Solution: Identify the original coordinates of the vertex: (10, 12) Reflection across the x-axis changes the sign of the y-coordinate while keeping the x-coordinate unchanged.
    Full step-by-step solution

    Step 1: Identify the original coordinates of the vertex: (10, 12) Step 2: Reflection across the x-axis changes the sign of the y-coordinate while keeping the x-coordinate unchanged. Step 3: Apply the transformation: new x-coordinate = 10, new y-coordinate = -12 Step 4: The reflected vertex is at (10, -12) The answer is (10, -12).

  2. (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.

  3. √(x² - 10x + 25) = 8 Answer: 13 Solution: Recognize that x² - 10x + 25 is a perfect square trinomial x² - 10x + 25 = (x - 5)² √((x - 5)²) = 8 The square root of a square gives the absolute value |x - 5| = 8 Case 1: x - 5 = 8 x = 8 + 5 x = 13 Case 2: x - 5 = -8 x = -8 + 5 x = -3 For x = 13: √(169 - 130 + 25) = √64 = 8 ✓ For x = -3: √(9 +…
    Full step-by-step solution

    Step 1: Recognize that x² - 10x + 25 is a perfect square trinomial x² - 10x + 25 = (x - 5)² Step 2: Substitute back into the equation √((x - 5)²) = 8 Step 3: The square root of a square gives the absolute value |x - 5| = 8 Step 4: Solve the absolute value equation Case 1: x - 5 = 8 x = 8 + 5 x = 13 Case 2: x - 5 = -8 x = -8 + 5 x = -3 Step 5: Check both solutions in the original equation For x = 13: √(169 - 130 + 25) = √64 = 8 ✓ For x = -3: √(9 + 30 + 25) = √64 = 8 ✓ Both solutions are valid, but the problem asks for the positive solution, so the answer is 13.

  4. Liam is designing a triangular garden plot using congruent triangles. He places triangle ABC with vertices at A(2, 3), B(5, 3), and C(2, 7). He wants to create a congruent triangle DEF by applying a sequence of rigid motions: first a translation 4 units right and 2 units down, then a reflection across the x-axis. What are the coordinates of the final triangle's vertices? Answer: D(6,-5), E(9,-5), F(6,-9) Solution: Rigid motions include translations, rotations, and reflections, all of which preserve distance and angle measures, maintaining congruence.
    Full step-by-step solution

    Rigid motions include translations, rotations, and reflections, all of which preserve distance and angle measures, maintaining congruence. When applying multiple transformations, the order matters - you apply them sequentially. Translations shift all points by the same amount, while reflections create mirror images across a line of reflection. The composition of rigid motions always results in another rigid motion.

  5. A triangle with vertices at (-6, 8), (9, -5), and (4, 11) is reflected across the x-axis. What are the coordinates of the reflected vertex that was originally at (9, -5)? Answer: (9, 5) Solution: Identify the original coordinates of the vertex: (9, -5) Reflection across the x-axis changes the sign of the y-coordinate while keeping the x-coordinate unchanged.
    Full step-by-step solution

    Step 1: Identify the original coordinates of the vertex: (9, -5) Step 2: Reflection across the x-axis changes the sign of the y-coordinate while keeping the x-coordinate unchanged. Step 3: Apply the transformation: new x-coordinate = 9, new y-coordinate = -(-5) = 5 Step 4: The reflected vertex coordinates are (9, 5) The answer is (9, 5).

  6. A triangle with vertices at (2, 3), (5, 7), and (8, 3) is reflected across the x-axis. What are the coordinates of the new vertices? Answer: (2, -3), (5, -7), (8, -3) Solution: The original vertices are (2, 3), (5, 7), and (8, 3). Reflection across the x-axis means we keep the x-coordinate the same and multiply the y-coordinate by -1. Reflecting (2, 3) gives (2, -3).
    Full step-by-step solution

    Step 1: The original vertices are (2, 3), (5, 7), and (8, 3). Step 2: Reflection across the x-axis means we keep the x-coordinate the same and multiply the y-coordinate by -1. Step 3: Reflecting (2, 3) gives (2, -3). Step 4: Reflecting (5, 7) gives (5, -7). Step 5: Reflecting (8, 3) gives (8, -3). Step 6: The new vertices after reflection are (2, -3), (5, -7), and (8, -3).

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

    Step 1: Start with the original point: (7, 5) Step 2: Apply reflection across y = x. This transformation swaps the x and y coordinates: (7, 5) becomes (5, 7) Step 3: Apply 90° counterclockwise rotation about the origin. This transformation follows the rule (x, y) → (-y, x): (5, 7) becomes (-7, 5) Step 4: Verify the transformations: Reflection: (7,5)→(5,7), Rotation: (5,7)→(-7,5) The final coordinates are (-7, 5)

  8. Liam is designing a triangular garden plot. He uses a coordinate grid to plan the layout, with the triangle having vertices at A(2, 1), B(5, 1), and C(3, 4). He wants to create a second identical plot by reflecting the triangle across the y-axis and then translating it 3 units down. What are the coordinates of the vertices of the second triangular garden plot after these two transformations? Answer: A'(-2, -2), B'(-5, -2), C'(-3, 1) Solution: A(2, 1) B(5, 1) C(3, 4) 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 triangle vertices** A(2, 1) B(5, 1) C(3, 4) --- **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, 1) → A_reflected(-2, 1) B(5, 1) → B_reflected(-5, 1) C(3, 4) → C_reflected(-3, 4) After reflection: A' = (-2, 1) B' = (-5, 1) C' = (-3, 4) --- **Step 3: Translation 3 units down** Translation 3 units down means subtract 3 from the y-coordinate. Rule: (x, y) → (x, y - 3) From reflected points: A_reflected(-2, 1) → A_final(-2, 1 - 3) = (-2, -2) B_reflected(-5, 1) → B_final(-5, 1 - 3) = (-5, -2) C_reflected(-3, 4) → C_final(-3, 4 - 3) = (-3, 1) --- **Step 4: Final coordinates** A'(-2, -2) B'(-5, -2) C'(-3, 1) --- **Final answer:** A'(-2, -2), B'(-5, -2), C'(-3, 1)