Question
On a coordinate plane, 2 right triangles are shown. The first triangle has points A (negative 1, 3), B (negative 1, 1), C (3, 1). The second triangle has points A prime (2, negative 2), B prime (2, negative 4), C prime (6, negative 4).
Which statements are true about triangle ABC and its translated image, A'B'C'? Select two options.
The rule for the translation can be written as T–5, 3(x, y).
The rule for the translation can be written as T3, –5(x, y).
The rule for the translation can be written as
(x, y) → (x + 3, y – 3).
The rule for the translation can be written as
(x, y) → (x – 3, y – 3).
Triangle ABC has been translated 3 units to the right and 5 units down.
Which statements are true about triangle ABC and its translated image, A'B'C'? Select two options.
The rule for the translation can be written as T–5, 3(x, y).
The rule for the translation can be written as T3, –5(x, y).
The rule for the translation can be written as
(x, y) → (x + 3, y – 3).
The rule for the translation can be written as
(x, y) → (x – 3, y – 3).
Triangle ABC has been translated 3 units to the right and 5 units down.
Answers
Answered by
GPT-4o mini
To determine the correct translation rule for the triangles and verify the truth of the statements, let's analyze the coordinates of the vertices of triangles ABC and A'B'C'.
**Triangle ABC has points:**
- A (-1, 3)
- B (-1, 1)
- C (3, 1)
**Triangle A'B'C' has points:**
- A' (2, -2)
- B' (2, -4)
- C' (6, -4)
Next, we can find the translation for each of the points from triangle ABC to triangle A'B'C'.
1. **For point A (-1, 3) to A' (2, -2)**:
- The x-coordinate changes from -1 to 2, which is a shift of \(2 - (-1) = 3\) units to the right.
- The y-coordinate changes from 3 to -2, which is a shift of \(-2 - 3 = -5\) units down.
2. **For point B (-1, 1) to B' (2, -4)**:
- The x-coordinate again changes from -1 to 2, which is a shift of 3 units to the right.
- The y-coordinate changes from 1 to -4, which is a shift of \(-4 - 1 = -5\) units down.
3. **For point C (3, 1) to C' (6, -4)**:
- The x-coordinate changes from 3 to 6, which is a shift of \(6 - 3 = 3\) units to the right.
- The y-coordinate changes from 1 to -4, which is a shift of \(-4 - 1 = -5\) units down.
From this analysis, the translation can be summarized as:
- Move 3 units to the right (add 3 to the x-coordinate).
- Move 5 units down (subtract 5 from the y-coordinate).
This can be expressed as the transformation:
\[
(x, y) \rightarrow (x + 3, y - 5).
\]
Now let's analyze the statements:
1. **The rule for the translation can be written as T–5, 3(x, y):**
- This translates to going down 5 and then right 3. This is NOT correct as per our analysis.
2. **The rule for the translation can be written as T3, –5(x, y):**
- This translates to going right 3 and down 5. This IS correct.
3. **The rule for the translation can be written as (x, y) → (x + 3, y – 3):**
- This suggests going right 3 and down 3. This is incorrect because we go down 5, not 3.
4. **The rule for the translation can be written as (x, y) → (x – 3, y – 3):**
- This suggests going left 3 and down 3. This is incorrect.
5. **Triangle ABC has been translated 3 units to the right and 5 units down:**
- This IS correct per our analysis.
Thus, the two true statements are:
- The rule for the translation can be written as T3, –5(x, y).
- Triangle ABC has been translated 3 units to the right and 5 units down.
**Triangle ABC has points:**
- A (-1, 3)
- B (-1, 1)
- C (3, 1)
**Triangle A'B'C' has points:**
- A' (2, -2)
- B' (2, -4)
- C' (6, -4)
Next, we can find the translation for each of the points from triangle ABC to triangle A'B'C'.
1. **For point A (-1, 3) to A' (2, -2)**:
- The x-coordinate changes from -1 to 2, which is a shift of \(2 - (-1) = 3\) units to the right.
- The y-coordinate changes from 3 to -2, which is a shift of \(-2 - 3 = -5\) units down.
2. **For point B (-1, 1) to B' (2, -4)**:
- The x-coordinate again changes from -1 to 2, which is a shift of 3 units to the right.
- The y-coordinate changes from 1 to -4, which is a shift of \(-4 - 1 = -5\) units down.
3. **For point C (3, 1) to C' (6, -4)**:
- The x-coordinate changes from 3 to 6, which is a shift of \(6 - 3 = 3\) units to the right.
- The y-coordinate changes from 1 to -4, which is a shift of \(-4 - 1 = -5\) units down.
From this analysis, the translation can be summarized as:
- Move 3 units to the right (add 3 to the x-coordinate).
- Move 5 units down (subtract 5 from the y-coordinate).
This can be expressed as the transformation:
\[
(x, y) \rightarrow (x + 3, y - 5).
\]
Now let's analyze the statements:
1. **The rule for the translation can be written as T–5, 3(x, y):**
- This translates to going down 5 and then right 3. This is NOT correct as per our analysis.
2. **The rule for the translation can be written as T3, –5(x, y):**
- This translates to going right 3 and down 5. This IS correct.
3. **The rule for the translation can be written as (x, y) → (x + 3, y – 3):**
- This suggests going right 3 and down 3. This is incorrect because we go down 5, not 3.
4. **The rule for the translation can be written as (x, y) → (x – 3, y – 3):**
- This suggests going left 3 and down 3. This is incorrect.
5. **Triangle ABC has been translated 3 units to the right and 5 units down:**
- This IS correct per our analysis.
Thus, the two true statements are:
- The rule for the translation can be written as T3, –5(x, y).
- Triangle ABC has been translated 3 units to the right and 5 units down.