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In 3–5 sentences, summarize the rotations that carry a regular pentagon onto itself.(4 points)

3 answers

A regular pentagon can be rotated about its center to coincide with itself at several specific angles. These rotations are 0 degrees (no rotation), 72 degrees, 144 degrees, 216 degrees, and 288 degrees. Each of these angles corresponds to a multiple of 72 degrees, which is the internal angle of rotation for a pentagon, reflecting the five symmetrical positions it can occupy in space. Consequently, these five rotations are the only transformations that will map the pentagon onto itself while preserving its shape and orientation.

Use the function rules to find the coordinates of ΔA"B"C" and predict the quadrant it will be in after the transformations. ΔABC , with coordinates A(−8,4) , B(−5,8) , and C(−3,3) , is transformed by (x,y)→(x+4,y−6) to produce ΔA′B′C′ . ΔA′B′C′ is transformed by the rule (x,y)→(x+5,y−3) . Create the rule for the composition transformation and predict what quadrant the image will be in.(4 points)

To find the coordinates of ΔA"B"C" after the given transformations, we first apply the transformations step-by-step.

  1. Transformation to ΔA'B'C': The transformation rule is \( (x, y) \rightarrow (x + 4, y - 6) \).

    • For point A(-8, 4):
      \( A' = (-8 + 4, 4 - 6) = (-4, -2) \)
    • For point B(-5, 8):
      \( B' = (-5 + 4, 8 - 6) = (-1, 2) \)
    • For point C(-3, 3):
      \( C' = (-3 + 4, 3 - 6) = (1, -3) \)

    Thus, the coordinates of ΔA'B'C' are:

    • A'(-4, -2)
    • B'(-1, 2)
    • C'(1, -3)

  2. Transformation to ΔA"B"C": The second transformation rule is \( (x, y) \rightarrow (x + 5, y - 3) \).

    • For point A'(-4, -2):
      \( A" = (-4 + 5, -2 - 3) = (1, -5) \)
    • For point B'(-1, 2):
      \( B" = (-1 + 5, 2 - 3) = (4, -1) \)
    • For point C'(1, -3):
      \( C" = (1 + 5, -3 - 3) = (6, -6) \)

    Thus, the coordinates of ΔA"B"C" are:

    • A"(1, -5)
    • B"(4, -1)
    • C"(6, -6)

  3. Composition Rule: To find the composition transformation directly, we can combine the two rules:

    • The first rule is \( (x, y) \rightarrow (x + 4, y - 6) \)
    • The second rule is \( (x, y) \rightarrow (x + 5, y - 3) \)

    Combining these, we get: \[ (x, y) \rightarrow (x + 4 + 5, y - 6 - 3) = (x + 9, y - 9) \]

  4. Quadrant Prediction: To predict the quadrant of ΔA"B"C", we analyze the final coordinates:

  • A"(1, -5): This point is in the fourth quadrant.
  • B"(4, -1): This point is also in the fourth quadrant.
  • C"(6, -6): This point is again in the fourth quadrant.

Since all three points are in the fourth quadrant, ΔA"B"C" will be located in the fourth quadrant after the transformations.