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Bayangan segitiga ABC dengan koordinat A(-3, 2), B(-2,2) dan C(2,2) ditranslasikan terhadap T[3, -4], kemudian direfleksikan terhadap sumbu x adalah....

Mathematics

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Answer

e

c. A′(0,2),B′(1,2) dan C′(5,2)\text{c. } A'(0, 2), B'(1, 2) \text{ dan } C'(5, 2)

Step-by-step solution

2 steps

To find the final coordinates of the triangle ABC after the given transformations, we will perform the transformations step-by-step.

The initial coordinates of the triangle are:
A(−3,2)A(-3, 2)
B(−2,2)B(-2, 2)
C(2,2)C(2, 2)

  1. Perform the translation.

    The triangle is translated by the vector T=[3−4]T = \begin{bmatrix} 3 \\ -4 \end{bmatrix}.
    A translation of a point (x,y)(x, y) by a vector [ab]\begin{bmatrix} a \\ b \end{bmatrix} results in the point (x+a,y+b)(x+a, y+b).

    Applying the translation to each vertex:
    For point A(−3,2)A(-3, 2):
    A′=(−3+3,2−4)=(0,−2)A' = (-3+3, 2-4) = (0, -2)

    For point B(−2,2)B(-2, 2):
    B′=(−2+3,2−4)=(1,−2)B' = (-2+3, 2-4) = (1, -2)

    For point C(2,2)C(2, 2):
    C′=(2+3,2−4)=(5,−2)C' = (2+3, 2-4) = (5, -2)

    So, after the translation, the coordinates are A′(0,−2)A'(0, -2), B′(1,−2)B'(1, -2), and C′(5,−2)C'(5, -2).

  2. Perform the reflection across the x-axis.

    The translated triangle is then reflected across the x-axis.
    A reflection of a point (x,y)(x, y) across the x-axis results in the point (x,−y)(x, -y).

    Applying the reflection to the translated vertices:
    For point A′(0,−2)A'(0, -2):
    A′′=(0,−(−2))=(0,2)A'' = (0, -(-2)) = (0, 2)

    For point B′(1,−2)B'(1, -2):
    B′′=(1,−(−2))=(1,2)B'' = (1, -(-2)) = (1, 2)

    For point C′(5,−2)C'(5, -2):
    C′′=(5,−(−2))=(5,2)C'' = (5, -(-2)) = (5, 2)

    The final coordinates of the image of triangle ABC are A′′(0,2)A''(0, 2), B′′(1,2)B''(1, 2), and C′′(5,2)C''(5, 2).

    Comparing these coordinates with the given options:
    a. A′(0,−2),B′(1,−2)A'(0, -2), B'(1, -2) dan C′(5,−2)C'(5, -2)
    b. A′(5,−2),B′(−1,−2)A'(5, -2), B'(-1, -2) dan C′(5,−2)C'(5, -2)
    c. A′(0,2),B′(1,2)A'(0, 2), B'(1, 2) dan C′(5,2)C'(5, 2)
    d. A′(0,−2),B′(1,−2)A'(0, -2), B'(1, -2) dan C′(−5,−2)C'(-5, -2)
    e. A′(3,−2),B′(1,2)A'(3, -2), B'(1, 2) dan C′(5,−2)C'(5, -2)

    The calculated final coordinates match option c.

    The final answer is c. A′(0,2),B′(1,2) dan C′(5,2)\boxed{\text{c. } A'(0, 2), B'(1, 2) \text{ dan } C'(5, 2)}.

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