Question

which series of transformations shows that triangle a is congruent to triangle b? triangle a triangle b a. reflect triangle a over the y - axis, rotate it 180° clockwise about the origin, and translate it 3 units up. b. reflect triangle a over the y - axis and translate it 2 units to the left and 8 units up. c. rotate triangle a 90° clockwise about (-1, -4), reflect it over the x - axis, and translate it 3 units to the left. d. rotate triangle a 180° about the point (-1, -4), reflect it over the x - axis, and translate it 3 units to the left.

Explanation:

Step1: Analyze option A

Reflecting over y - axis changes sign of x - coordinates. Rotating 180° clockwise about origin changes signs of both x and y coordinates. Translating 3 units up adds 3 to y - coordinates. This sequence does not map triangle A to triangle B.

Step2: Analyze option B

Reflecting triangle A over the y - axis changes the sign of the x - coordinates of its vertices. Then translating 2 units to the left (subtract 2 from x - coordinates) and 8 units up (add 8 to y - coordinates).
Let's assume a vertex of triangle A is (x,y). After reflection over y - axis, it becomes (-x,y). After translation, it becomes (-x - 2,y + 8). This sequence maps triangle A to triangle B.

Step3: Analyze option C

Rotating 90° clockwise about a point and then reflecting over x - axis and translating left does not map triangle A to triangle B.

Step4: Analyze option D

Rotating 180° about a point, reflecting over x - axis and translating left does not map triangle A to triangle B.

Answer:

B. Reflect triangle A over the y - axis and translate it 2 units to the left and 8 units up.