Find the area of the triangle with vertices `P(4,5),Q(4,-2) and R(-6,2).

Options

A. 21 sq. units
B. 35 sq, units
C. 20 sq. units
D. 40 sq. units

Answer

To find the area of the triangle with vertices \( P(4,5) \), \( Q(4,-2) \), and \( R(-6,2) \), 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:
- \( (x_1, y_1) = (4, 5) \) for point \( P \)
- \( (x_2, y_2) = (4, -2) \) for point \( Q \)
- \( (x_3, y_3) = (-6, 2) \) for point \( R \)

### Step 1: Substitute the coordinates into the formula

Substituting the coordinates into the area formula:

\[ \text{Area} = \frac{1}{2} \left| 4((-2) - 2) + 4(2 - 5) + (-6)(5 - (-2)) \right| \]

### Step 2: Simplify the expression

Calculating each term:

1. \( 4((-2) - 2) = 4(-4) = -16 \)
2. \( 4(2 - 5) = 4(-3) = -12 \)
3. \( -6(5 - (-2)) = -6(5 + 2) = -6(7) = -42 \)

Now, substituting these values back into the area formula:

\[ \text{Area} = \frac{1}{2} \left| -16 - 12 - 42 \right| \]

### Step 3: Combine the terms

Combining the terms inside the absolute value:

\[ -16 - 12 - 42 = -70 \]

### Step 4: Calculate the absolute value

Now, taking the absolute value:

\[ \left| -70 \right| = 70 \]

### Step 5: Final calculation of the area

Finally, calculating the area:

\[ \text{Area} = \frac{1}{2} \times 70 = 35 \]

Thus, the area of the triangle is \( 35 \) square units.

### Final Answer:
The area of the triangle is \( 35 \) square units.

---