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Congruence Concepts

Grade 8 · Geometry · Worksheet 3

  1. A triangle with vertices at (8, 5), (12, 9), and (6, 13) is reflected across the y-axis. What are the coordinates of the reflected vertex that was originally at (12, 9)? Answer: ______________
  2. Liam is designing a triangular logo for his robotics team. He creates triangle ABC with vertices at A(2,1), B(5,1), and C(3,4). To test different placements, he performs a sequence of rigid motions: first a translation 3 units right and 2 units up, then a reflection across the x-axis, and finally a 90° counterclockwise rotation about the origin. What are the coordinates of the final position of vertex C after all these transformations? Answer: ______________
  3. 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: ______________
  4. A triangle with vertices at (4, 9), (11, 6), and (7, 14) is reflected across the x-axis. What are the coordinates of the reflected vertex that was originally at (11, 6)? Answer: ______________
  5. 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: ______________
  6. A triangle with vertices at (4, 9), (8, 13), and (2, 17) is reflected across the x-axis. What are the coordinates of the reflected vertex that was originally at (8, 13)? Answer: ______________
  7. Aroha draws a triangle on a coordinate plane with vertices at A(10, 12), B(18, 12), and C(10, 20). She then reflects the triangle across the line y = x, and then translates the reflected triangle 9 units to the left and 4 units down. What are the coordinates of the final position of vertex C after both transformations? Answer: ______________
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Answer Key & Explanations

Congruence Concepts · Grade 8 · Worksheet 3

  1. A triangle with vertices at (8, 5), (12, 9), and (6, 13) is reflected across the y-axis. What are the coordinates of the reflected vertex that was originally at (12, 9)? Answer: (-12, 9) Solution: Identify the original coordinates of the vertex: (12, 9) 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: (12, 9) 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 = -12, new y-coordinate = 9 Step 4: The reflected vertex coordinates are (-12, 9) The answer is (-12, 9).

  2. Liam is designing a triangular logo for his robotics team. He creates triangle ABC with vertices at A(2,1), B(5,1), and C(3,4). To test different placements, he performs a sequence of rigid motions: first a translation 3 units right and 2 units up, then a reflection across the x-axis, and finally a 90° counterclockwise rotation about the origin. What are the coordinates of the final position of vertex C after all these transformations? Answer: (5,-1) Solution: Rigid motions include translations, rotations, and reflections, which preserve distances and angles between points. When applying multiple transformations, the order matters significantly.
    Full step-by-step solution

    Rigid motions include translations, rotations, and reflections, which preserve distances and angles between points. When applying multiple transformations, the order matters significantly. A good strategy is to track one point through each transformation step by step, using the rules for each type of motion.

  3. 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.

  4. A triangle with vertices at (4, 9), (11, 6), and (7, 14) is reflected across the x-axis. What are the coordinates of the reflected vertex that was originally at (11, 6)? Answer: (11, -6) Solution: Identify the original coordinates of the vertex: (11, 6) 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: (11, 6) 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 = 11, new y-coordinate = -6 Step 4: The reflected vertex coordinates are (11, -6) The answer is (11, -6).

  5. 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)

  6. A triangle with vertices at (4, 9), (8, 13), and (2, 17) is reflected across the x-axis. What are the coordinates of the reflected vertex that was originally at (8, 13)? Answer: (8, -13) Solution: Identify the original coordinates of the vertex: (8, 13) 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: (8, 13) 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 = 8, new y-coordinate = -13 Step 4: The reflected vertex coordinates are (8, -13) The answer is (8, -13).

  7. Aroha draws a triangle on a coordinate plane with vertices at A(10, 12), B(18, 12), and C(10, 20). She then reflects the triangle across the line y = x, and then translates the reflected triangle 9 units to the left and 4 units down. What are the coordinates of the final position of vertex C after both transformations? Answer: (11, 6) Solution: Start with the original coordinates of vertex C: C(10, 20). The rule for this reflection is (x, y) → (y, x). So C(10, 20) becomes C'(20, 10).
    Full step-by-step solution

    Step 1: Start with the original coordinates of vertex C: C(10, 20). Step 2: Apply the first transformation - reflection across the line y = x. The rule for this reflection is (x, y) → (y, x). So C(10, 20) becomes C'(20, 10). Step 3: Apply the second transformation - translation 9 units left and 4 units down. A translation left subtracts from the x-coordinate, and a translation down subtracts from the y-coordinate. So the rule is (x, y) → (x - 9, y - 4). Step 4: Apply the translation to C'(20, 10): new x = 20 - 9 = 11, new y = 10 - 4 = 6. So the final coordinates are (11, 6). The answer is (11, 6).