Final Answer:
Figure 1 and Figure 2 are congruent. The congruence transformation is a rotation of 180° about the origin, followed by a translation 3 units down.
──────────────────────────────────────
Explanation:
Let’s look at the two triangles in the coordinate plane:
- Figure 1 has vertices at approximately: (−2, 1), (−1, −1), and (−3, −1).
- Figure 2 has vertices at approximately: (2, −4), (1, −2), and (3, −2).
We want to see if we can move Figure 1 (using only rotations, reflections, or translations) to exactly match Figure 2.
Step 1: Try rotating Figure 1
180° around the origin.
When you rotate a point (x, y) 180° about the origin, it becomes (−x, −y).
So:
- (−2, 1) → (2, −1)
- (−1, −1) → (1, 1)
- (−3, −1) → (3, 1)
That gives us a triangle at (2, −1), (1, 1), (3, 1) — still not Figure 2.
But wait — look again! The problem says *“Figure 1 can be mapped onto Figure 2 by a rotation 180° counterclockwise about the origin and a translation 3 units down.”* Let’s test that full sequence:
After 180° rotation:
(−2, 1) → (2, −1)
Then translate
down 3 units: subtract 3 from y → (2, −4)
✔ matches Figure 2’s first vertex.
(−1, −1) → (1, 1) → (1, −2)
✔ matches second vertex.
(−3, −1) → (3, 1) → (3, −2)
✔ matches third vertex.
All three points line up perfectly with Figure 2.
So yes — they are congruent, and the transformation is:
rotate 180° about the origin, then
translate 3 units down.
Parent Tip: Review the logic above to help your child master the concept of congruent transformations worksheet.
