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Rectangle ABCD has coordinates A(-10, 5) , B(10, 5), C(10, 0), and D(-10, 0) . Rectangle A'B'C'D' has coordinates A'(-10, -5), B' (10, -5) . C°(10, 0), and D'(-10, 0). Rectangle A" B" C" D" has coordinates A" (-2, - B"(2, B"(2, - 1), C" (2, 0) , and D"(-2, 0) .

Which transformations describe why rectangles ABCD and A" B" C" D" are similar?
Rectangle ABCD was dilated by a scale factor of - and then rotated 90° counterclockwise to form
Rectangle A " B" C" D"
Rectangle ABCD was reflected across the x-axis and then dilated by a scale factor of ¿ to form Rectangle A " B" C" D"
Rectangle ABCD was reflected across the y-axis and then dilated by a scale factor of 5 to form Rectangle A " B" C" D"
Rectangle ABCD was dilated by a scale factor of 5 and then rotated 90° counterclockwise to form
Rectangle A " B" C" D"

1 answer

To determine the transformations that describe why rectangles ABCD and A''B''C''D'' are similar, let's examine the properties and transformations involved.

  1. Dimensions of Rectangle ABCD:

    • Length: From A to B = 10 - (-10) = 20
    • Width: From D to A = 5 - 0 = 5
    • Therefore, the dimensions of rectangle ABCD are 20 (length) x 5 (width).

  2. Dimensions of Rectangle A''B''C''D'':

    • From A'' to C'' you can see that the horizontal distance is from -2 to 2 (or 2 - (-2) = 4).
    • From A'' to D'' the vertical distance is from -1 to 0 (or 0 - (-1) = 1).
    • Therefore, the dimensions of rectangle A''B''C''D'' are 4 (length) x 1 (width).

  3. Checking similarity:

    • For two rectangles to be similar, the ratios of their corresponding lengths must be the same.
    • The ratio of the length of ABCD to that of A''B''C''D'' is 20:4 which simplifies to 5:1.
    • The ratio of the width of ABCD to that of A''B''C''D'' is 5:1 which is also 5:1.

Since both lengths and widths have the same ratio of 5:1, rectangles ABCD and A''B''C''D'' are similar.

Now, let's analyze the transformations provided in the options:

  1. Rectangle ABCD was dilated by a scale factor of -1 and then rotated 90° counterclockwise to form Rectangle A''B''C''D'':

    • This would not maintain the dimensions.

  2. Rectangle ABCD was reflected across the x-axis and then dilated by a scale factor of ¼ to form Rectangle A''B''C''D'':

    • Reflecting across the x-axis would change the sign of y-coordinates but not x-coordinates and dilating by ¼ would make the dimensions small enough (20 becomes 5; 5 becomes 1.25), which does not match.

  3. Rectangle ABCD was reflected across the y-axis and then dilated by a scale factor of 5 to form Rectangle A''B''C''D'':

    • This option does not match because dilating by a scale factor of 5 would increase the dimensions, not decrease them.

  4. Rectangle ABCD was dilated by a scale factor of ¼ and then rotated 90° counterclockwise to form Rectangle A''B''C''D'':

    • This option would reduce the size correctly and maintains the aspect ratio resulting in the right coordinates.

From the options provided, the correct transformation that describes why rectangles ABCD and A''B''C''D'' are similar is: Rectangle ABCD was dilated by a scale factor of ¼ and then rotated 90° counterclockwise to form Rectangle A''B''C''D''.