the refection coordinates of the point (8,-4) which are translated by T and then continued by reflection on the line y=x are (5, -3). t translation is

To solve for the translation \( T \), we know that the point \( (8, -4) \) undergoes two transformations:

1. A translation \( T \).
2. A reflection over the line \( y = x \), resulting in the final point \( (5, -3) \).

Let's break this into two parts:

### Step 1: Reflection over the line \( y = x \)
The reflection of a point \( (x, y) \) over the line \( y = x \) swaps the coordinates, turning \( (x, y) \) into \( (y, x) \).

Let the point after translation be \( (x', y') \). If \( (x', y') \) is reflected over the line \( y = x \), the new point will be \( (y', x') \), and we know this equals \( (5, -3) \). Therefore:

\[
y' = 5 \quad \text{and} \quad x' = -3
\]

So, after translation, the point becomes \( (-3, 5) \).

### Step 2: Finding the translation \( T \)
Translation \( T \) moves the point \( (8, -4) \) to \( (-3, 5) \). The translation can be expressed as:

\[
T = (x' - x, y' - y)
\]

Substitute the coordinates of the original point \( (8, -4) \) and the point after translation \( (-3, 5) \):

\[
T = (-3 - 8, 5 - (-4)) = (-11, 9)
\]

### Final Answer:
The translation \( T \) is \( (-11, 9) \).

Would you like more details or have any questions? Here are 5 related questions to consider:

1. What is the reflection of a point over the line \( y = -x \)?
2. How do you calculate the translation for a general point \( (x, y) \)?
3. Can you reflect a point over a vertical or horizontal line instead of \( y = x \)?
4. How would this problem change if the final point was different?
5. What would happen if you applied two reflections over different lines?

**Tip:** Always swap the coordinates when reflecting a point over the line \( y = x \).

The refection coordinates of the point (8,-4) which are translated by T and then continued by reflection on the line y=x are (5, -3). t translation is

This problem involves finding the translation and reflection of the point (8, -4) over the line y=x, resulting in the coordinates (5, -3). Learn how to calculate the translation and apply reflection using key geometric formulas.

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