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Congruence Concepts

Grade 8 · Geometry · Worksheet 1

  1. A triangle has vertices at A(1, 2), B(4, 2), and C(1, 5) on a coordinate plane. The triangle is first rotated 90° counterclockwise about the origin, then reflected across the x-axis. What are the coordinates of the final position of vertex C after both transformations? Answer: ______________
  2. Liam is designing a triangular logo for his robotics team. He creates triangle ABC with vertices at A(2, 3), B(5, 1), and C(7, 4). He wants to create a congruent triangle DEF by applying a sequence of rigid motions. First, he reflects triangle ABC across the y-axis, then rotates the resulting triangle 90° counterclockwise about the origin, and finally translates it 3 units right and 2 units up. What are the coordinates of vertex D in the final triangle DEF? Answer: ______________
  3. A triangle with vertices at (4, -2), (9, 6), and (-3, 8) is reflected across the x-axis. What are the coordinates of the reflected vertex that was originally at (9, 6)? Answer: ______________
  4. A triangle has vertices at points A(2, 1), B(5, 1), and C(2, 4) on a coordinate plane. The triangle is reflected across the y-axis, then translated 3 units to the right. What are the coordinates of the final position of vertex C? Answer: ______________
  5. √(x² - 8x + 16) = 12 Answer: ______________
  6. 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: ______________
  7. A triangle with vertices at (-3, 8), (6, -4), and (5, 11) is reflected across the y-axis. What are the coordinates of the reflected vertex that was originally at (6, -4)? Answer: ______________
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Answer Key & Explanations

Congruence Concepts · Grade 8 · Worksheet 1

  1. A triangle has vertices at A(1, 2), B(4, 2), and C(1, 5) on a coordinate plane. The triangle is first rotated 90° counterclockwise about the origin, then reflected across the x-axis. What are the coordinates of the final position of vertex C after both transformations? Answer: (5,1) Solution: Start with the original coordinates of point C: C(1, 5) Apply the first transformation - rotation 90° counterclockwise about the origin. The rule for this rotation is (x, y) → (-y, x).
    Full step-by-step solution

    Step 1: Start with the original coordinates of point C: C(1, 5) Step 2: Apply the first transformation - rotation 90° counterclockwise about the origin. The rule for this rotation is (x, y) → (-y, x). For C(1, 5): x = 1, y = 5 After rotation: (-5, 1) Step 3: Apply the second transformation - reflection across the x-axis. The rule for this reflection is (x, y) → (x, -y). For the point (-5, 1): x = -5, y = 1 After reflection: (-5, -1) Step 4: Verify the final coordinates: (-5, -1) The answer is (-5, -1).

  2. Liam is designing a triangular logo for his robotics team. He creates triangle ABC with vertices at A(2, 3), B(5, 1), and C(7, 4). He wants to create a congruent triangle DEF by applying a sequence of rigid motions. First, he reflects triangle ABC across the y-axis, then rotates the resulting triangle 90° counterclockwise about the origin, and finally translates it 3 units right and 2 units up. What are the coordinates of vertex D in the final triangle DEF? Answer: (1, 5) Solution: Rigid motions are transformations that preserve distance and shape, including reflections, rotations, and translations. When applying multiple transformations, we work step by step from the original coordinates. Each transformation follows specific rules: reflection across the y-axis changes…
    Full step-by-step solution

    Rigid motions are transformations that preserve distance and shape, including reflections, rotations, and translations. When applying multiple transformations, we work step by step from the original coordinates. Each transformation follows specific rules: reflection across the y-axis changes (x,y) to (-x,y); 90° counterclockwise rotation about the origin changes (x,y) to (-y,x); and translation adds the specified values to the coordinates. The final position represents the image after all transformations have been applied.

  3. A triangle with vertices at (4, -2), (9, 6), and (-3, 8) is reflected across the x-axis. What are the coordinates of the reflected vertex that was originally at (9, 6)? Answer: (9, -6) Solution: Identify the original coordinates of the vertex: (9, 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: (9, 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 = 9, new y-coordinate = -6 Step 4: The reflected vertex coordinates are (9, -6) The answer is (9, -6).

  4. A triangle has vertices at points A(2, 1), B(5, 1), and C(2, 4) on a coordinate plane. The triangle is reflected across the y-axis, then translated 3 units to the right. What are the coordinates of the final position of vertex C? Answer: (1, 4) Solution: C = (2, 4) Reflecting 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: Identify the original coordinates of vertex C** From the problem: C = (2, 4) --- **Step 2: Reflect across the y-axis** Reflecting across the y-axis changes the sign of the x-coordinate, while the y-coordinate stays the same. Rule: (x, y) → (-x, y) Apply to C(2, 4): New x = -2 New y = 4 So after reflection: C' = (-2, 4) --- **Step 3: Translate 3 units to the right** Translating right means adding 3 to the x-coordinate; y-coordinate stays the same. Rule: (x, y) → (x + 3, y) Apply to C'(-2, 4): New x = -2 + 3 = 1 New y = 4 So after translation: C'' = (1, 4) --- **Final answer:** (1, 4)

  5. √(x² - 8x + 16) = 12 Answer: 16 Solution: Recognize that x² - 8x + 16 is a perfect square trinomial. It factors as (x - 4)². Rewrite the equation: √((x - 4)²) = 12.
    Full step-by-step solution

    Step 1: Recognize that x² - 8x + 16 is a perfect square trinomial. It factors as (x - 4)². Step 2: Rewrite the equation: √((x - 4)²) = 12. Step 3: The square root of a square is the absolute value: |x - 4| = 12. Step 4: Solve the absolute value equation. This gives two cases: Case 1: x - 4 = 12 → x = 12 + 4 → x = 16 Case 2: x - 4 = -12 → x = -12 + 4 → x = -8 Step 5: Check both solutions in the original equation: For x = 16: √(16² - 8*16 + 16) = √(256 - 128 + 16) = √144 = 12 (Valid) For x = -8: √((-8)² - 8*(-8) + 16) = √(64 + 64 + 16) = √144 = 12 (Also valid) Since the problem asks for a solution and both are mathematically valid, the positive solution x = 16 is typically preferred in this context. The answer is 16.

  6. 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.

  7. A triangle with vertices at (-3, 8), (6, -4), and (5, 11) is reflected across the y-axis. What are the coordinates of the reflected vertex that was originally at (6, -4)? Answer: (-6, -4) Solution: Identify the original coordinates of the vertex: (6, -4) 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: (6, -4) 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 = -6, new y-coordinate = -4 Step 4: The reflected vertex coordinates are (-6, -4) The answer is (-6, -4).