Question

which two series of transformations would carry triangle abc to triangle pqr? a. a translation 2 units right followed by a translation 2 units down b. a translation 6 units right followed by a translation 2 units down c. a 180° rotation about point b followed by a translation 2 units right and 2 units down d. a reflection across a horizontal line 1 unit below (overline{bc}) followed by a rotation 180° about point b e. a reflection across a horizontal line 1 unit below (overline{bc}) followed by a reflection across a vertical line 1 unit right of point b

Explanation:

Step1: Analyze horizontal translation

Count the horizontal grid - units between corresponding points of $\triangle ABC$ and $\triangle PQR$. For example, the horizontal distance between point $B$ and point $Q$. We find that the horizontal translation is 6 units to the right.

Step2: Analyze vertical translation

Count the vertical grid - units between corresponding points. For example, the vertical distance between the new position of a point after the horizontal translation and its corresponding point in $\triangle PQR$. We find that the vertical translation is 2 units down.

Step3: Check other options

For option C, a $180^{\circ}$ rotation about point $B$ will change the orientation in a way that is not consistent with the given triangles. For option D and E, reflections and rotations will change the orientation and position in a way that does not match the transformation from $\triangle ABC$ to $\triangle PQR$.

Answer:

B. a translation 6 units right followed by a translation 2 units down