Find the area of `Delta PQR`, whose vertices are P (-5, 7) , Q (-4, -5) and R (4, 5) .

Answer

To find the area of triangle PQR with vertices P(-5, 7), Q(-4, -5), and R(4, 5), we can use the formula for the area of a triangle given its vertices in the coordinate plane. The formula is:

\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

### Step-by-step Solution:

1. **Identify the coordinates of the vertices:**
- Let \( P(x_1, y_1) = (-5, 7) \)
- Let \( Q(x_2, y_2) = (-4, -5) \)
- Let \( R(x_3, y_3) = (4, 5) \)

2. **Substitute the coordinates into the area formula:**
\[ \text{Area} = \frac{1}{2} \left| -5(-5 - 5) + (-4)(5 - 7) + 4(7 - (-5)) \right| \]

3. **Calculate each term inside the absolute value:**
- First term:
\[ -5(-5 - 5) = -5(-10) = 50 \]
- Second term:
\[ -4(5 - 7) = -4(-2) = 8 \]
- Third term:
\[ 4(7 - (-5)) = 4(7 + 5) = 4(12) = 48 \]

4. **Combine the terms:**
\[ 50 + 8 + 48 = 106 \]

5. **Calculate the area:**
\[ \text{Area} = \frac{1}{2} \left| 106 \right| = \frac{106}{2} = 53 \]

### Final Answer:
The area of triangle PQR is \( 53 \) square units.

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