1. the image of abc is a'b'c. what transformations would result in this image? (photo 1) 2. abc is dilated about the origin. what scale factor was used to make the image a'b'c? (photo 2) 3.a'b'c'd' is the image of abcd. what transformation(s) would result in this image? (photo 3)

#1) Reflected across the line y=x; #2) -2; #3) Reflected across the x-axis and then the y-axis, reflected across the y-axis and then the x-axis, and rotated 180 degrees around the origin.

Step-by-step explanation:

#1) The pre-image point A(5, -5) is mapped to A'(-5, 5).  The x- and y-coordinates were switched.  The only one of the transformations listed that would accomplish this is reflection across the line y=x.

If we reflect across the y-axis, (x, y)→(-x, y).  Rotating -90 degrees around the origin would then map (-x, y)→(-y, -x).  This is not the same as switching the x- and y-coordinates.

If we rotate -90 degrees around the origin, this maps (x, y)→(-y, x).  Reflecting across the x-axis would then map (-y, x)→(-y, -x).  This is not the same as switching the x- and y-coordinates.

If we rotate 90 degrees around the origin, this maps (x, y)→(y, -x).  Reflecting across the y-axis then maps (y, -x)→(-y, -x).  This is not the same as switching the x- and y-coordinates.

#2) We can see in the pre-image that the length of BC is 1.  We can also see that the length of B'C' is 2; however it has been reflected as well, so it has been multiplied by a scale factor of -2.

#3) The x- and y-coordinates of the pre-image ABCD have been negated to form A'B'C'D'.  This can be done by reflecting across both the y- and x-axes, no matter which order; this means the first and third options are correct.

This can also be done through a rotation 180 degrees about the origin, as it maps every point (x, y)→(-x, -y).

1. The second choice is appropriate.
  ΔABC is reflected over the line y = x.

2. The scale factor is -2, but the premise is incorrect. The figure is not dilated about the origin. It is dilated about the point (0, 2/3).

3. All but the second choice will describe the transformation.
• reflected across x then y
• reflected across y then x
• rotated 180° about the origin

-4(2x-10)=-x+61
-8x+40=-x+61
-7x+40=61
-7x=21
x=-3
the number is -3