Use the image to answer the question. Translate triangle ABC by 3 units to the left and 5 units down. Which of the following are the coordinates of new triangle A′B′C′? (1 point) Responses A′(1,10), B′(−1,6), C′(4,7) upper A prime left parenthesis 1 comma 10 right parenthesis , upper B prime left parenthesis negative 1 comma 6 right parenthesis , upper C prime left parenthesis 4 comma 7 right parenthesis A′(1,0), B′(−1,−4), C′(4,−2) upper A prime left parenthesis 1 comma 0 right parenthesis , upper B prime left parenthesis negative 1 comma negative 4 right parenthesis , upper C prime left parenthesis 4 comma negative 2 right parenthesis A′(7,0), B′(5,−4), C′(10,−2) upper A prime left parenthesis 7 comma 0 right parenthesis , upper B prime left parenthesis 5 comma negative 4 right parenthesis , upper C prime left parenthesis 10 comma negative 2 right parenthesis A′(−1,2), B′(−3,−2), C′(2,0)

To translate triangle ABC by 3 units to the left and 5 units down, we apply the following transformations to the coordinates of each vertex:

1. **Translate 3 units to the left**: This means we will subtract 3 from the x-coordinate of each vertex.
2. **Translate 5 units down**: This means we will subtract 5 from the y-coordinate of each vertex.

If we represent the original coordinates of points A, B, and C as \( A(x_A, y_A), B(x_B, y_B), C(x_C, y_C) \), the new coordinates will be:

- \( A'(x_A - 3, y_A - 5) \)
- \( B'(x_B - 3, y_B - 5) \)
- \( C'(x_C - 3, y_C - 5) \)

Since we don't have the original coordinates of triangle ABC given in the problem, we can't directly compute the new coordinates. However, we can check the provided answer choices to see which set of coordinates could result from this transformation.

Let's analyze the four response options step by step, assuming some hypothetical original coordinates for triangle ABC:

1. **Response**: A′(1,10), B′(−1,6), C′(4,7)
- Notably, this transformation does not involve any subtraction by 3 or 5.

2. **Response**: A′(1,0), B′(−1,−4), C′(4,−2)
- For point A': (1 + 3, 0 + 5) → (4, 5) is not plausible; others do not appear to follow the translation either.

3. **Response**: A′(7,0), B′(5,−4), C′(10,−2)
- Similar to the previous responses, the transformations do not follow the rules as well.

4. **Response**: A′(−1,2), B′(−3,−2), C′(2,0)
- For point A', (−1 + 3, 2 + 5) → (2, 7); this doesn’t match either.

Since we lack the original coordinates necessary for verifying each transformation specific to ABC, the right answer cannot be concluded from this information alone.

It appears there are no coordinate sets provided that plausibly fit the specified translation of 3 left and 5 down without knowing the original points. If you have any coordinate points for triangle ABC, I can proceed with the calculations accordingly!

If coordinates (−2,−5) undergo a translation vector of <2,−3> in the coordinate plane, what would the resulting coordinates be?(1 point) Responses (0,−2) left parenthesis 0 comma negative 2 right parenthesis (4,2) left parenthesis 4 comma 2 right parenthesis (4,−2) left parenthesis 4 comma negative 2 right parenthesis (0,−8)

To find the resulting coordinates after applying the translation vector \( \langle 2, -3 \rangle \) to the original coordinates \( (-2, -5) \), we need to follow these steps:

1. **Add the x-component of the translation vector to the x-coordinate:**
\[
x' = -2 + 2 = 0
\]

2. **Add the y-component of the translation vector to the y-coordinate:**
\[
y' = -5 + (-3) = -5 - 3 = -8
\]

So, after translating the point \( (-2, -5) \) by the vector \( \langle 2, -3 \rangle \), the new coordinates will be \( (0, -8) \).

Given the response options, the correct choice is:
- \( (0, -8) \)

se the images to answer the question. If rectangle ABDC is translated horizontally −1 and vertically +3, which of the following correctly depicts rectangle A′B′D′C′? (1 point) Responses Image with alt text: A rectangle is labeled upper A prime upper B prime upper D prime upper C prime. The area inside of the rectangle is shaded, and each of the vertices is labeled. Image with alt text: A rectangle is formed by connecting four data points. Image with alt text: A quadrilateral is labeled upper A prime upper B prime upper D prime upper C prime. Each of the four corners of the shape are labeled with the coordinates.

To determine which image depicts the translated rectangle A′B′D′C′ after rectangle ABDC has been translated horizontally by -1 unit and vertically by +3 units, we need to follow these transformation steps:

1. **Horizontal Translation (-1)**: This means we will subtract 1 from the x-coordinates of all vertices.
2. **Vertical Translation (+3)**: This means we will add 3 to the y-coordinates of all vertices.

If we denote the original coordinates of A, B, C, and D as \( A(x_A, y_A), B(x_B, y_B), C(x_C, y_C), D(x_D, y_D) \), then the new coordinates after translation will be:

- \( A'(x_A - 1, y_A + 3) \)
- \( B'(x_B - 1, y_B + 3) \)
- \( D'(x_D - 1, y_D + 3) \)
- \( C'(x_C - 1, y_C + 3) \)

Since we don’t have the original coordinates of rectangle ABDC and can’t directly calculate the new coordinates, we can't specifically identify the correct image. However, you would be looking for an image where:

- Each vertex A′, B′, C′, and D′ is shifted 1 unit to the left and 3 units up from their respective original positions.

In practice, check the provided images for rectangles where the positions of the vertices correspond to the described transformation. Ensure that every vertex has been moved as specified when comparing images.

If you can describe the coordinates from the images provided, I’d be happy to help further with the analysis!