The point P (2, -4) is reflected about the line x = 0 to get the image Q. The point Q is reflected about the line y = 0 to get the image R.
Find the area of figure PQR.

Answer

To solve the problem, we need to follow these steps:

### Step 1: Reflect Point P about the line x = 0
The coordinates of point P are (2, -4). When we reflect a point across the line x = 0 (the y-axis), the x-coordinate changes sign while the y-coordinate remains the same.

- Original point P: (2, -4)
- Reflected point Q: (-2, -4)

### Step 2: Reflect Point Q about the line y = 0
Now, we reflect point Q (-2, -4) across the line y = 0 (the x-axis). When reflecting across the x-axis, the y-coordinate changes sign while the x-coordinate remains the same.

- Original point Q: (-2, -4)
- Reflected point R: (-2, 4)

### Step 3: Identify the coordinates of points P, Q, and R
Now we have the coordinates of all three points:
- P(2, -4)
- Q(-2, -4)
- R(-2, 4)

### Step 4: Calculate the area of triangle PQR
To find the area of triangle PQR, we can use the formula for the area of a triangle given by the coordinates of its vertices:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Where:
- \( (x_1, y_1) = P(2, -4) \)
- \( (x_2, y_2) = Q(-2, -4) \)
- \( (x_3, y_3) = R(-2, 4) \)

Substituting the coordinates into the formula:
\[ \text{Area} = \frac{1}{2} \left| 2(-4 - 4) + (-2)(4 + 4) + (-2)(-4 + 4) \right| \]
Calculating each term:
1. \( 2(-4 - 4) = 2(-8) = -16 \)
2. \( -2(4 + 4) = -2(8) = -16 \)
3. \( -2(-4 + 4) = -2(0) = 0 \)

Now, substituting back into the area formula:
\[ \text{Area} = \frac{1}{2} \left| -16 - 16 + 0 \right| = \frac{1}{2} \left| -32 \right| = \frac{1}{2} \times 32 = 16 \]

Thus, the area of triangle PQR is \( 16 \) square units.

### Final Answer
The area of figure PQR is \( 16 \) square units.

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