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Transformations

Grade 8 · Geometry · Worksheet 3

  1. Triangle Hana with vertices H(4,8), A(6,8), N(4,12) is reflected over the x-axis. What are the coordinates of the image? Answer: ______________
  2. Triangle ABC with vertices A(2, 7), B(5, 7), C(2, 3) is reflected over the y-axis. What are the coordinates of A'? Answer: ______________
  3. Charlotte is designing a banner for a school event. She starts with a rectangle that is 15 cm by 10 cm. She then applies a dilation to the rectangle using a scale factor of 3/2. After the dilation, she reflects the new rectangle over the x-axis. Which statement correctly describes the relationship between the original rectangle and the final transformed rectangle?
    • A. They are similar but not congruent.
    • B. They are congruent.
    • C. They are neither similar nor congruent.
    • D. They are identical in size and orientation.
  4. Triangle Emma with vertices E(5,10), M(10,15), M(15,10) is reflected over the y-axis, then rotated 90° counterclockwise about the origin. What are the coordinates of the final image? Answer: ______________
  5. Triangle Isabella with vertices I(2,7), S(7,12), A(12,7) is reflected over the x-axis. What are the coordinates of I'? Answer: ______________
  6. Triangle Isabella has vertices I(8,9), S(12,9), and A(8,13). It is reflected over the x-axis, then rotated 90° counterclockwise about the origin. What are the coordinates of the final image of vertex A?
    • A. (13,8)
    • B. (13,-8)
    • C. (-13,-8)
    • D. (-8,-13)
  7. Matiu is designing a geometric pattern using transformations. He starts with a triangle at coordinates A(2,4), B(6,4), and C(4,8). He performs a 180° rotation about the origin, followed by a translation 4 units to the right. What is the x-coordinate of the final position of point C? Answer: ______________
  8. Triangle ABC with vertices A(2,7), B(7,2), C(12,7) is reflected over the x-axis. What are the coordinates of A'B'C'? Answer: ______________
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Answer Key & Explanations

Transformations · Grade 8 · Worksheet 3

  1. Triangle Hana with vertices H(4,8), A(6,8), N(4,12) is reflected over the x-axis. What are the coordinates of the image? Answer: H'(4,-8), A'(6,-8), N'(4,-12) Solution: A reflection over the x-axis is a transformation that flips a figure across the x-axis.
    Full step-by-step solution

    A reflection over the x-axis is a transformation that flips a figure across the x-axis. This transformation preserves the x-coordinates but changes the sign of the y-coordinates, creating a mirror image across the horizontal axis. The resulting figure is congruent to the original.

  2. Triangle ABC with vertices A(2, 7), B(5, 7), C(2, 3) is reflected over the y-axis. What are the coordinates of A'? Answer: (-2, 7) Solution: The original coordinates of point A are (2, 7). When reflecting over the y-axis, the x-coordinate changes sign. The new x-coordinate becomes -2.
    Full step-by-step solution

    Step 1: The original coordinates of point A are (2, 7). Step 2: When reflecting over the y-axis, the x-coordinate changes sign. Step 3: The new x-coordinate becomes -2. Step 4: The y-coordinate remains the same: 7. Step 5: Therefore, the coordinates of A' are (-2, 7).

  3. Charlotte is designing a banner for a school event. She starts with a rectangle that is 15 cm by 10 cm. She then applies a dilation to the rectangle using a scale factor of 3/2. After the dilation, she reflects the new rectangle over the x-axis. Which statement correctly describes the relationship between the original rectangle and the final transformed rectangle? Answer: A. They are similar but not congruent. Solution: When a figure undergoes a dilation, its size changes but its shape remains the same, making the original and dilated figures similar.
    Full step-by-step solution

    When a figure undergoes a dilation, its size changes but its shape remains the same, making the original and dilated figures similar. A reflection is a rigid motion that preserves both size and shape, so it does not affect similarity. Therefore, after a dilation and a reflection, the original and final figures are similar but not congruent because their sizes are different. For example, if you have a triangle and you enlarge it by a scale factor greater than 1, and then flip it, the triangles are still similar because their angles match and side lengths are proportional, but they are not congruent due to the size change.

  4. Triangle Emma with vertices E(5,10), M(10,15), M(15,10) is reflected over the y-axis, then rotated 90° counterclockwise about the origin. What are the coordinates of the final image? Answer: E'(-10,-5), M'(-15,-10), M'(-10,-15) Solution: When performing multiple transformations, apply them in sequence. Reflection over the y-axis preserves y-values but negates x-values. A 90° counterclockwise rotation about the origin swaps coordinates and changes signs in a specific pattern.
  5. Triangle Isabella with vertices I(2,7), S(7,12), A(12,7) is reflected over the x-axis. What are the coordinates of I'? Answer: (2,-7) Solution: The original point I has coordinates (2,7) When reflecting over the x-axis, the x-coordinate remains the same: 2 The y-coordinate changes sign: 7 becomes -7 Therefore, the reflected point I' has coordinates (2,-7) The answer is (2,-7).
    Full step-by-step solution

    Step 1: The original point I has coordinates (2,7) Step 2: When reflecting over the x-axis, the x-coordinate remains the same: 2 Step 3: The y-coordinate changes sign: 7 becomes -7 Step 4: Therefore, the reflected point I' has coordinates (2,-7) The answer is (2,-7).

  6. Triangle Isabella has vertices I(8,9), S(12,9), and A(8,13). It is reflected over the x-axis, then rotated 90° counterclockwise about the origin. What are the coordinates of the final image of vertex A? Answer: B. (13,-8) Solution: Start with the original coordinates of vertex A: (8,13) A(8,13) → A'(8,-13) Apply 90° counterclockwise rotation about the origin: (x,y) → (-y,x) A'(8,-13) → A''(-(-13),8) = (13,8) The final coordinates of vertex A after both transformations are (13,8).
    Full step-by-step solution

    Step 1: Start with the original coordinates of vertex A: (8,13) Step 2: Apply reflection over the x-axis: (x,y) → (x,-y) A(8,13) → A'(8,-13) Step 3: Apply 90° counterclockwise rotation about the origin: (x,y) → (-y,x) A'(8,-13) → A''(-(-13),8) = (13,8) The final coordinates of vertex A after both transformations are (13,8).

  7. Matiu is designing a geometric pattern using transformations. He starts with a triangle at coordinates A(2,4), B(6,4), and C(4,8). He performs a 180° rotation about the origin, followed by a translation 4 units to the right. What is the x-coordinate of the final position of point C? Answer: 0 Solution: Original coordinates of point C are (4,8). After 180° rotation about the origin, the coordinates become (-4,-8). After translating 4 units to the right, add 4 to the x-coordinate: -4 + 4 = 0.
    Full step-by-step solution

    Step 1: Original coordinates of point C are (4,8). Step 2: After 180° rotation about the origin, the coordinates become (-4,-8). Step 3: After translating 4 units to the right, add 4 to the x-coordinate: -4 + 4 = 0. Step 4: The y-coordinate remains -8, but the question asks for the x-coordinate only. The final x-coordinate of point C is 0.

  8. Triangle ABC with vertices A(2,7), B(7,2), C(12,7) is reflected over the x-axis. What are the coordinates of A'B'C'? Answer: A(2,-7), B(7,-2), C(12,-7) Solution: A reflection over the x-axis is a transformation that flips a figure across the x-axis. The x-coordinates remain unchanged, while the y-coordinates become their opposites.
    Full step-by-step solution

    A reflection over the x-axis is a transformation that flips a figure across the x-axis. The x-coordinates remain unchanged, while the y-coordinates become their opposites. This transformation preserves the shape and size of the figure, making the original and reflected triangles congruent.