the center of the reflected image lie? \( (4,3) \)

To determine where the **center of a reflected image** lies, it's essential to know two key pieces of information: 1. **The Original Center:** The coordinates of the original object's center before reflection. 2. **The Reflection Axis or Point:** The line or point across which the object is being reflected. Given your mention of the point \( (4, 3) \), I'll assume you're either: - **Reflecting an object so that its new center is at \( (4, 3) \)**, - **Or, determining if \( (4, 3) \) is the center after reflection.** Since the exact nature of the reflection (across a line or a point) isn't specified, I'll outline the common scenarios below: --- ### 1. **Reflection Across a Line** **a. Reflection Across the X-Axis** - **Original Center:** \( (a, b) \) - **Reflected Center:** \( (a, -b) \) *Example:* If the original center is \( (4, 3) \), reflecting across the x-axis would place the new center at \( (4, -3) \). --- **b. Reflection Across the Y-Axis** - **Original Center:** \( (a, b) \) - **Reflected Center:** \( (-a, b) \) *Example:* Reflecting \( (4, 3) \) across the y-axis results in \( (-4, 3) \). --- **c. Reflection Across the Line \( y = x \)** - **Original Center:** \( (a, b) \) - **Reflected Center:** \( (b, a) \) *Example:* Reflecting \( (4, 3) \) across \( y = x \) yields \( (3, 4) \). --- ### 2. **Reflection Across a Point** When reflecting across a **specific point** (let's say point \( C(h, k) \)): - **Original Center:** \( (a, b) \) - **Reflected Center:** \( (2h - a, 2k - b) \) *Example:* If you're reflecting \( (4, 3) \) across the origin \( (0, 0) \), the new center becomes \( (-4, -3) \). --- ### 3. **Determining the Reflection Center** If you know the **original center** and the **reflected center**, you can find the **axis or point of reflection**. For instance: - **For Reflection Across a Point:** Suppose the original center is \( (x_1, y_1) \), and the reflected center is \( (4, 3) \). The center of reflection \( C(h, k) \) can be found using: \[ h = \frac{x_1 + 4}{2}, \quad k = \frac{y_1 + 3}{2} \] - **For Reflection Across a Line:** You'd set up equations based on the type of reflection (e.g., across the x-axis, y-axis, or another line) to find the relationship between the original and reflected centers. --- ### 4. **Example Problem** **Problem:** Suppose you have a circle with its center at \( (2, 5) \). You reflect this circle across the y-axis. **Question:** What is the center of the reflected image? **Solution:** Reflection across the y-axis changes the sign of the x-coordinate. - Original Center: \( (2, 5) \) - Reflected Center: \( (-2, 5) \) --- ### Conclusion Without specific details about the **original center** or the **reflection method**, it's challenging to provide a definitive answer. However, if you intended to state that after reflection, the center lies at \( (4, 3) \), then based on the reflection type, you can determine the original center or the reflection axis accordingly using the methods outlined above. If you provide more details about the original position and the reflection criteria, I can offer a more precise explanation!