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Transformations

Grade 8 · Geometry · Worksheet 2

  1. Kaia is designing a geometric pattern using transformations. She starts with a triangle with vertices at (12, 4), (16, 4), and (12, 9). She applies a dilation centered at the origin with a scale factor of 2.5. What is the y-coordinate of the image of vertex (12, 9) after this dilation? Answer: ______________
  2. Triangle Aroha has vertices at A(9,10), B(11,10), and C(9,14). It is reflected over the x-axis. What are the coordinates of the image?
    • A. A'(10,9), B'(10,11), C'(14,9)
    • B. A'(9,10), B'(11,10), C'(9,14)
    • C. A'(-9,10), B'(-11,10), C'(-9,14)
    • D. A'(9,-10), B'(11,-10), C'(9,-14)
  3. Isabella is designing a triangular garden plot with vertices at (8, 12), (14, 12), and (14, 18). She creates a similar garden by applying a dilation centered at the origin with a scale factor of 1.5. What is the y-coordinate of the dilated image of the vertex that was originally at (14, 18)? Answer: ______________
  4. Triangle Aroha with vertices A(3,5), B(7,5), C(5,9) is reflected over the x-axis. What are the coordinates of A'B'C'? Answer: ______________
  5. Triangle Emma with vertices E(5,10), M(10,10), M(5,15) is reflected over the x-axis. What are the coordinates of E'? Answer: ______________
  6. Sophia is designing a logo for a tech company using a triangular shape. She starts with triangle DEF with vertices at D(1, 1), E(6, 1), and F(1, 6). She applies a 90° counterclockwise rotation about the origin, followed by a dilation with scale factor 2 centered at the origin. What are the final coordinates of vertex F after both transformations? Answer: ______________
  7. Aroha is designing a geometric pattern using transformations. She starts with triangle PQR with vertices at (1, 3), (5, 3), and (3, 7). She performs two transformations: first a 90° counterclockwise rotation about the origin, then a dilation with scale factor 3 centered at the origin. Which statement correctly describes the relationship between the original triangle and the final image?
    • A. They are congruent
    • B. They are neither similar nor congruent
    • C. They are identical
    • D. They are similar but not congruent
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Answer Key & Explanations

Transformations · Grade 8 · Worksheet 2

  1. Kaia is designing a geometric pattern using transformations. She starts with a triangle with vertices at (12, 4), (16, 4), and (12, 9). She applies a dilation centered at the origin with a scale factor of 2.5. What is the y-coordinate of the image of vertex (12, 9) after this dilation? Answer: 22.5 Solution: Identify the vertex we're transforming: (12, 9) The dilation is centered at the origin with scale factor 2.5 To find the image coordinates, multiply both x and y coordinates by 2.5 Calculate the y-coordinate: 9 × 2.5 = 22.5 The y-coordinate of the image is 22.5
    Full step-by-step solution

    Step 1: Identify the vertex we're transforming: (12, 9) Step 2: The dilation is centered at the origin with scale factor 2.5 Step 3: To find the image coordinates, multiply both x and y coordinates by 2.5 Step 4: Calculate the y-coordinate: 9 × 2.5 = 22.5 Step 5: The y-coordinate of the image is 22.5

  2. Triangle Aroha has vertices at A(9,10), B(11,10), and C(9,14). It is reflected over the x-axis. What are the coordinates of the image? Answer: D. A'(9,-10), B'(11,-10), C'(9,-14) Solution: Reflection over the x-axis means the x-coordinate stays the same and the y-coordinate changes sign. Original point A(9,10) becomes A'(9,-10) Original point B(11,10) becomes B'(11,-10) Original point C(9,14) becomes C'(9,-14) The image vertices are A'(9,-10), B'(11,-10), and C'(9,-14) The correct…
    Full step-by-step solution

    Step 1: Reflection over the x-axis means the x-coordinate stays the same and the y-coordinate changes sign. Step 2: Original point A(9,10) becomes A'(9,-10) Step 3: Original point B(11,10) becomes B'(11,-10) Step 4: Original point C(9,14) becomes C'(9,-14) Step 5: The image vertices are A'(9,-10), B'(11,-10), and C'(9,-14) The correct answer is A'(9,-10), B'(11,-10), C'(9,-14).

  3. Isabella is designing a triangular garden plot with vertices at (8, 12), (14, 12), and (14, 18). She creates a similar garden by applying a dilation centered at the origin with a scale factor of 1.5. What is the y-coordinate of the dilated image of the vertex that was originally at (14, 18)? Answer: 27 Solution: Identify the vertex to be dilated: (14, 18) Apply the dilation with scale factor 1.5 centered at the origin Multiply the y-coordinate by the scale factor: 18 × 1.5 = 27 The y-coordinate of the dilated vertex is 27
    Full step-by-step solution

    Step 1: Identify the vertex to be dilated: (14, 18) Step 2: Apply the dilation with scale factor 1.5 centered at the origin Step 3: Multiply the y-coordinate by the scale factor: 18 × 1.5 = 27 Step 4: The y-coordinate of the dilated vertex is 27

  4. Triangle Aroha with vertices A(3,5), B(7,5), C(5,9) is reflected over the x-axis. What are the coordinates of A'B'C'? Answer: (3,-5),(7,-5),(5,-9) Solution: Original vertices are A(3,5), B(7,5), C(5,9) Reflection over x-axis: x-coordinates remain the same, y-coordinates change sign A' = (3,-5) B' = (7,-5) C' = (5,-9) The coordinates of A'B'C' are (3,-5), (7,-5), (5,-9)
    Full step-by-step solution

    Step 1: Original vertices are A(3,5), B(7,5), C(5,9) Step 2: Reflection over x-axis: x-coordinates remain the same, y-coordinates change sign Step 3: A' = (3,-5) Step 4: B' = (7,-5) Step 5: C' = (5,-9) Step 6: The coordinates of A'B'C' are (3,-5), (7,-5), (5,-9)

  5. Triangle Emma with vertices E(5,10), M(10,10), M(5,15) is reflected over the x-axis. What are the coordinates of E'? Answer: (5,-10) Solution: Identify the original coordinates of point E: (5,10) Apply reflection over the x-axis - keep x-coordinate the same, change sign of y-coordinate x-coordinate remains: 5 y-coordinate changes from 10 to -10 The reflected point E' is (5,-10)
    Full step-by-step solution

    Step 1: Identify the original coordinates of point E: (5,10) Step 2: Apply reflection over the x-axis - keep x-coordinate the same, change sign of y-coordinate Step 3: x-coordinate remains: 5 Step 4: y-coordinate changes from 10 to -10 Step 5: The reflected point E' is (5,-10)

  6. Sophia is designing a logo for a tech company using a triangular shape. She starts with triangle DEF with vertices at D(1, 1), E(6, 1), and F(1, 6). She applies a 90° counterclockwise rotation about the origin, followed by a dilation with scale factor 2 centered at the origin. What are the final coordinates of vertex F after both transformations? Answer: (-12, 2) Solution: A rotation is a rigid motion that turns a figure around a fixed point without changing its size or shape. A 90° counterclockwise rotation about the origin changes the coordinates of any point (x, y) to (-y, x).
    Full step-by-step solution

    A rotation is a rigid motion that turns a figure around a fixed point without changing its size or shape. A 90° counterclockwise rotation about the origin changes the coordinates of any point (x, y) to (-y, x). A dilation is a transformation that changes the size of a figure but not its shape, and when centered at the origin, it multiplies each coordinate by the scale factor. When performing multiple transformations, the order matters, and you apply them sequentially to find the final image.

  7. Aroha is designing a geometric pattern using transformations. She starts with triangle PQR with vertices at (1, 3), (5, 3), and (3, 7). She performs two transformations: first a 90° counterclockwise rotation about the origin, then a dilation with scale factor 3 centered at the origin. Which statement correctly describes the relationship between the original triangle and the final image? Answer: D. They are similar but not congruent Solution: Analyze the first transformation - 90° counterclockwise rotation about the origin. This is a rigid motion that preserves both size and shape, so the rotated triangle is congruent to the original.
    Full step-by-step solution

    Step 1: Analyze the first transformation - 90° counterclockwise rotation about the origin. This is a rigid motion that preserves both size and shape, so the rotated triangle is congruent to the original. Step 2: Analyze the second transformation - dilation with scale factor 3 centered at the origin. A dilation preserves shape but changes size by the scale factor, so all side lengths are multiplied by 3. Step 3: Determine the overall effect. The composition of a rotation (rigid motion) and a dilation (with scale factor not equal to 1) results in a similarity transformation. The final image has the same shape as the original but all side lengths are 3 times longer. Step 4: Since the scale factor is 3 (not 1), the triangles are not congruent. They are similar because they have the same shape but different sizes. The correct answer is They are similar but not congruent.