The equation of one side of a rectangle is `3x-4y -10=0` and the coordinates of two of its vertices are `(-2,1) and (2, 4)`. Then, the area of the rectangle is

Options

A. 20
B. 40
C. 10
D. 30

Answer

To find the area of the rectangle given the equation of one side and the coordinates of two vertices, we can follow these steps:

### Step 1: Verify the Points
We need to check if the given points (-2, 1) and (2, 4) lie on the line represented by the equation \(3x - 4y - 10 = 0\).

**Calculation for Point (-2, 1):**
\[ 3(-2) - 4(1) - 10 = -6 - 4 - 10 = -20 \quad (\text{not equal to } 0) \]

**Calculation for Point (2, 4):**
\[ 3(2) - 4(4) - 10 = 6 - 16 - 10 = -20 \quad (\text{not equal to } 0) \]

Since both points do not satisfy the equation, they lie outside the line.

### Step 2: Calculate the Length of the Rectangle
The length of the rectangle can be calculated using the distance formula between the two points (-2, 1) and (2, 4).

**Distance Formula:**
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the points:
\[ d = \sqrt{(2 - (-2))^2 + (4 - 1)^2} = \sqrt{(2 + 2)^2 + (4 - 1)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]
So, the length of the rectangle is 5.

### Step 3: Calculate the Width of the Rectangle
The width of the rectangle is the perpendicular distance from one of the points to the line \(3x - 4y - 10 = 0\).

**Perpendicular Distance Formula:**
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
Where \(A = 3\), \(B = -4\), \(C = -10\), and we can use either point. Let's use (-2, 1).

Substituting into the formula:
\[ d = \frac{|3(-2) - 4(1) - 10|}{\sqrt{3^2 + (-4)^2}} = \frac{|-6 - 4 - 10|}{\sqrt{9 + 16}} = \frac{|-20|}{\sqrt{25}} = \frac{20}{5} = 4 \]
So, the width of the rectangle is 4.

### Step 4: Calculate the Area of the Rectangle
The area \(A\) of the rectangle is given by:
\[ A = \text{Length} \times \text{Width} = 5 \times 4 = 20 \]

### Final Answer
The area of the rectangle is \(20\).

---