Worksheet 1Worksheet 2Worksheet 3

The download includes this worksheet followed by the answer key with explanations. In the print dialog choose “Save as PDF” to download.

lessonbunny.com
Name: ______________________________ Date: ______________

Congruence Concepts

Grade 8 · Geometry · Worksheet 3

  1. Liam is designing a triangular garden plot. He places stakes at points A(2, 3), B(5, 7), and C(8, 3). His friend Noah wants to create a congruent triangular plot adjacent to it by applying a sequence of rigid motions: first a translation 4 units right and 2 units up, then a reflection across the vertical line x=9. What are the coordinates of the vertices of Noah's triangular garden plot after applying these transformations? Answer: ______________
  2. 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: ______________
  3. 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: ______________
  4. (2x + 3)² - (x - 4)² = ? Answer: ______________
  5. 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). To create a matching logo, he applies a sequence of rigid motions: first a translation 3 units left and 2 units down, then a reflection across the x-axis. What are the coordinates of the vertices of the final triangle? Answer: ______________
  6. Liam is designing a triangular logo for his robotics team. He uses a coordinate grid to position the triangle with vertices at A(2, 1), B(5, 1), and C(3, 4). To create a symmetrical design, he reflects the triangle across the y-axis to get triangle A'B'C'. Then, he rotates this new triangle 90 degrees counterclockwise about the origin to get triangle A''B''C''. What are the coordinates of vertex C'' in the final position? Answer: ______________
  7. Aroha is creating a geometric pattern using congruent triangles on a coordinate grid. She draws triangle ABC with vertices at A(-3, 5), B(1, 5), and C(-1, 1). To complete her design, she reflects triangle ABC across the y-axis to get triangle A'B'C', then rotates triangle A'B'C' 90 degrees clockwise about the origin to get triangle A''B''C''. What are the coordinates of vertex A'' in the final position? Answer: ______________
lessonbunny.com

Answer Key & Explanations

Congruence Concepts · Grade 8 · Worksheet 3

  1. Liam is designing a triangular garden plot. He places stakes at points A(2, 3), B(5, 7), and C(8, 3). His friend Noah wants to create a congruent triangular plot adjacent to it by applying a sequence of rigid motions: first a translation 4 units right and 2 units up, then a reflection across the vertical line x=9. What are the coordinates of the vertices of Noah's triangular garden plot after applying these transformations? Answer: (10, 9), (8, 5), (6, 9) Solution: Rigid motions are transformations that preserve the size and shape of geometric figures, including translations, reflections, and rotations.
    Full step-by-step solution

    Rigid motions are transformations that preserve the size and shape of geometric figures, including translations, reflections, and rotations. When applying multiple transformations in sequence, each transformation affects all points of the figure. Translations move all points by the same distance in the same direction, while reflections create mirror images across a line of reflection. The order of transformations matters, as different sequences can produce different final positions even with the same individual transformations.

  2. 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).

  3. 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).

  4. (2x + 3)² - (x - 4)² = ? Answer: 3x² + 20x - 7 Solution: Expand (2x + 3)² (2x + 3)² = (2x)² + 2(2x)(3) + 3² = 4x² + 12x + 9 Expand (x - 4)² (x - 4)² = x² - 2(x)(4) + (-4)² = x² - 8x + 16 (4x² + 12x + 9) - (x² - 8x + 16) = 4x² + 12x + 9 - x² + 8x - 16 4x² - x² = 3x² 12x + 8x = 20x 9 - 16 = -7 3x² + 20x - 7 The answer is 3x² + 20x - 7.
    Full step-by-step solution

    Step 1: Expand (2x + 3)² (2x + 3)² = (2x)² + 2(2x)(3) + 3² = 4x² + 12x + 9 Step 2: Expand (x - 4)² (x - 4)² = x² - 2(x)(4) + (-4)² = x² - 8x + 16 Step 3: Subtract the second expansion from the first (4x² + 12x + 9) - (x² - 8x + 16) = 4x² + 12x + 9 - x² + 8x - 16 Step 4: Combine like terms 4x² - x² = 3x² 12x + 8x = 20x 9 - 16 = -7 Step 5: Write the final expression 3x² + 20x - 7 The answer is 3x² + 20x - 7.

  5. 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). To create a matching logo, he applies a sequence of rigid motions: first a translation 3 units left and 2 units down, then a reflection across the x-axis. What are the coordinates of the vertices of the final triangle? Answer: A'(-1, -5), B'(2, -3), C'(4, -6) Solution: Rigid motions preserve the size and shape of geometric figures. Translations move every point the same distance in the same direction, while reflections create mirror images.
    Full step-by-step solution

    Rigid motions preserve the size and shape of geometric figures. Translations move every point the same distance in the same direction, while reflections create mirror images. When applying multiple transformations, the order matters - you apply them sequentially from the first to the last. The resulting figure will be congruent to the original, just in a different position or orientation.

  6. Liam is designing a triangular logo for his robotics team. He uses a coordinate grid to position the triangle with vertices at A(2, 1), B(5, 1), and C(3, 4). To create a symmetrical design, he reflects the triangle across the y-axis to get triangle A'B'C'. Then, he rotates this new triangle 90 degrees counterclockwise about the origin to get triangle A''B''C''. What are the coordinates of vertex C'' in the final position? Answer: (1, -3) Solution: Rigid motions are transformations that preserve the distance and shape of a figure. A reflection flips a figure over a line, creating a mirror image. A rotation turns a figure around a fixed point.
    Full step-by-step solution

    Rigid motions are transformations that preserve the distance and shape of a figure. A reflection flips a figure over a line, creating a mirror image. A rotation turns a figure around a fixed point. When performing multiple transformations, you apply them in sequence, using the result of the first transformation as the starting point for the next. The final coordinates depend on the specific order of these operations.

  7. Aroha is creating a geometric pattern using congruent triangles on a coordinate grid. She draws triangle ABC with vertices at A(-3, 5), B(1, 5), and C(-1, 1). To complete her design, she reflects triangle ABC across the y-axis to get triangle A'B'C', then rotates triangle A'B'C' 90 degrees clockwise about the origin to get triangle A''B''C''. What are the coordinates of vertex A'' in the final position? Answer: (5, 3) Solution: Start with point A(-3, 5). Reflect across the y-axis. The x-coordinate changes sign, the y-coordinate stays the same.
    Full step-by-step solution

    Step 1: Start with point A(-3, 5). Step 2: Reflect across the y-axis. The x-coordinate changes sign, the y-coordinate stays the same. So A' = (3, 5). Step 3: Rotate A' 90 degrees clockwise about the origin. For a 90-degree clockwise rotation, (x, y) becomes (y, -x). So A'' = (5, -3). Step 4: The final coordinates of vertex A'' are (5, -3).