Question
Two triangles, upper A upper B upper C and upper A prime upper B prime upper C prime, are graphed on a coordinate plane. The x-axis ranges from negative 7 to 9 in increments of 1. The y-axis ranges from negative 3 to 8 in increments of 1.
Maria determined that △ABC is congruent with △A′B′C′ because it was rotated 270° (counterclockwise). Do you agree with every part of Maria’s conclusion?
Statement #1: Yes. △ABC was rotated 270° (counterclockwise) and is congruent with △A′B′C′ .
Statement #2: No. △ABC is congruent with △A′B′C′ , but it was rotated 90° (counterclockwise), not 270° .
Statement #3: No. △ABC was rotated −90° (clockwise) and is not congruent with △A′B′C′ .
Maria determined that △ABC is congruent with △A′B′C′ because it was rotated 270° (counterclockwise). Do you agree with every part of Maria’s conclusion?
Statement #1: Yes. △ABC was rotated 270° (counterclockwise) and is congruent with △A′B′C′ .
Statement #2: No. △ABC is congruent with △A′B′C′ , but it was rotated 90° (counterclockwise), not 270° .
Statement #3: No. △ABC was rotated −90° (clockwise) and is not congruent with △A′B′C′ .
Answers
Answered by
GPT-4o mini
To determine whether Maria's conclusion is correct, we need to examine the statements carefully.
1. **Understanding the rotations:**
- A 270° counterclockwise rotation is equivalent to a 90° clockwise rotation. This means if a triangle is rotated 270° counterclockwise, it will end up in the same position as if it were rotated 90° clockwise.
2. **Congruence of triangles:**
- Two triangles are said to be congruent if they have the same size and shape, regardless of their orientation. If triangle \( \triangle ABC \) is congruent to triangle \( \triangle A'B'C' \), they can indeed be congruent if one is a rotation of the other.
Now, let's analyze each statement:
- **Statement #1:** "Yes. \( \triangle ABC \) was rotated 270° (counterclockwise) and is congruent with \( \triangle A'B'C' \)."
- This statement can be incorrect because while congruence may hold, the specific description of rotation (270° counterclockwise) may not be correct.
- **Statement #2:** "No. \( \triangle ABC \) is congruent with \( \triangle A'B'C' \), but it was rotated 90° (counterclockwise), not 270°."
- If \( \triangle ABC \) is indeed rotated 90° counterclockwise to match \( \triangle A'B'C' \), this statement would be correct.
- **Statement #3:** "No. \( \triangle ABC \) was rotated -90° (clockwise) and is not congruent with \( \triangle A'B'C' \)."
- If \( \triangle ABC \) is not congruent to \( \triangle A'B'C' \) and was rotated -90° clockwise, then this statement would be correct. However, that would contradict the assertion that they are congruent.
Based on this analysis, **Statement #2** is the most accurate. It acknowledges the congruence of the triangles while correcting Maria's claim about the specific rotation degree and direction. Thus, I would agree with Statement #2.
1. **Understanding the rotations:**
- A 270° counterclockwise rotation is equivalent to a 90° clockwise rotation. This means if a triangle is rotated 270° counterclockwise, it will end up in the same position as if it were rotated 90° clockwise.
2. **Congruence of triangles:**
- Two triangles are said to be congruent if they have the same size and shape, regardless of their orientation. If triangle \( \triangle ABC \) is congruent to triangle \( \triangle A'B'C' \), they can indeed be congruent if one is a rotation of the other.
Now, let's analyze each statement:
- **Statement #1:** "Yes. \( \triangle ABC \) was rotated 270° (counterclockwise) and is congruent with \( \triangle A'B'C' \)."
- This statement can be incorrect because while congruence may hold, the specific description of rotation (270° counterclockwise) may not be correct.
- **Statement #2:** "No. \( \triangle ABC \) is congruent with \( \triangle A'B'C' \), but it was rotated 90° (counterclockwise), not 270°."
- If \( \triangle ABC \) is indeed rotated 90° counterclockwise to match \( \triangle A'B'C' \), this statement would be correct.
- **Statement #3:** "No. \( \triangle ABC \) was rotated -90° (clockwise) and is not congruent with \( \triangle A'B'C' \)."
- If \( \triangle ABC \) is not congruent to \( \triangle A'B'C' \) and was rotated -90° clockwise, then this statement would be correct. However, that would contradict the assertion that they are congruent.
Based on this analysis, **Statement #2** is the most accurate. It acknowledges the congruence of the triangles while correcting Maria's claim about the specific rotation degree and direction. Thus, I would agree with Statement #2.
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