Question

trapezoid efgh has coordinates e(-3, 4), f(1, 4), g(3, 1), and h(-5, 1). trapezoid efgh has coordinates e(-12, 16), f(4, 16), g(12, 4), and h(-20, 4). trapezoid efgh has coordinates e(12, -16), f(-4, -16), g(-12, -4), and h(20, -4).

which transformations describe why trapezoids efgh and efgh are similar?

• trapezoid efgh was translated 4 units right and 4 units up and then rotated 180° clockwise.

• trapezoid efgh was dilated by a scale factor of 1/4 and then reflected across the x-axis.

• trapezoid efgh was rotated 90° clockwise and then dilated by a scale factor of 4.

• trapezoid efgh was dilated by a scale factor of 4 and then rotated 180° counterclockwise.

Explanation:

Step1: Analyze dilation from EFGH to E'F'G'H'

First, check the coordinates of E(-3,4) to E'(-12,16). The x - coordinate: \(-3\times4=-12\), the y - coordinate: \(4\times4 = 16\). Similarly, for F(1,4) to F'(4,16): \(1\times4 = 4\), \(4\times4=16\); G(3,1) to G'(12,4): \(3\times4 = 12\), \(1\times4 = 4\); H(-5,1) to H'(-20,4): \(-5\times4=-20\), \(1\times4 = 4\). So the dilation factor from EFGH to E'F'G'H' is 4.

Step2: Analyze rotation from E'F'G'H' to E''F''G''H''

A \(180^{\circ}\) counter - clockwise (or clockwise) rotation about the origin has the rule \((x,y)\to(-x,-y)\). For E'(-12,16), applying \(180^{\circ}\) rotation: \((-(-12),-16)=(12,-16)\) which is E''. For F'(4,16): \((-4,-16)\) which is F''. For G'(12,4): \((-12,-4)\) which is G''. For H'(-20,4): \((20,-4)\) which is H''. So from E'F'G'H' to E''F''G''H'', it is a \(180^{\circ}\) counter - clockwise (or clockwise) rotation. Combining the two transformations: first dilation by scale factor 4, then rotation by \(180^{\circ}\) counter - clockwise.

Now check the other options:

• Option 1: The translation and rotation do not match the coordinate changes.
• Option 2: The scale factor is wrong (it should be 4, not \(\frac{1}{4}\)).
• Option 3: The rotation (90° clockwise) and dilation order and factor are wrong.

Answer:

Trapezoid EFGH was dilated by a scale factor of 4 and then rotated \(180^{\circ}\) counterclockwise.