The image of (4,5,-3) in the X-Y plane is (4,5,3)

Answer

To determine whether the statement "The image of (4, 5, -3) in the X-Y plane is (4, 5, 3)" is true or false, we can follow these steps:

### Step 1: Understand the Coordinates
The given point is \( P(4, 5, -3) \). The coordinates consist of:
- \( x = 4 \)
- \( y = 5 \)
- \( z = -3 \)

### Step 2: Identify the X-Y Plane
The X-Y plane is defined by the equation \( z = 0 \). Any point on this plane has a z-coordinate of 0.

### Step 3: Calculate the Distance from the Point to the X-Y Plane
The distance from point \( P(4, 5, -3) \) to the X-Y plane can be calculated by taking the absolute value of the z-coordinate:
\[ \text{Distance} = |z| = |-3| = 3 \]
This means that the point \( P \) is 3 units below the X-Y plane.

### Step 4: Determine the Coordinates of the Foot of the Perpendicular
The foot of the perpendicular from point \( P \) to the X-Y plane will have the same x and y coordinates as point \( P \), but the z-coordinate will be 0:
\[ \text{Foot of perpendicular} = (4, 5, 0) \]

### Step 5: Find the Image of the Point
The image of the point in the X-Y plane can be found by reflecting it across the plane. Since the distance from the point to the plane is 3 units, the image will be 3 units above the X-Y plane. Therefore, we add 3 to the z-coordinate of the foot of the perpendicular:
\[ \text{Image} = (4, 5, 0 + 3) = (4, 5, 3) \]

### Step 6: Conclusion
Since we have found that the image of the point \( (4, 5, -3) \) in the X-Y plane is indeed \( (4, 5, 3) \), the statement is true.

### Final Answer
The statement is **True**.

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