One side of a rectangle lies along the line 4x + 7y + 5 = 0. Two of its vertices are (-3, 1) and (1, 1). Which of the following is not an equation of the other three straight lines?

Options

A. 14x - 8y = 6
B. 7x - 4y = -25
C. 4 x + 7y = 11
D. 14 -8y = - 25

Answer

To solve the problem step by step, we need to determine which of the given equations is not a valid equation of the other three sides of the rectangle. We know that one side of the rectangle lies along the line given by the equation \(4x + 7y + 5 = 0\) and two vertices of the rectangle are given as \((-3, 1)\) and \((1, 1)\).

### Step 1: Identify the line along which one side of the rectangle lies.
The equation of the line is given as:
\[ 4x + 7y + 5 = 0 \]
This can be rewritten in slope-intercept form \(y = mx + b\) to find its slope.

### Step 2: Find the slope of the line.
Rearranging the equation:
\[ 7y = -4x - 5 \implies y = -\frac{4}{7}x - \frac{5}{7} \]
The slope \(m\) of this line is \(-\frac{4}{7}\).

### Step 3: Determine the coordinates of the vertices.
The two vertices given are:
- \(A(-3, 1)\)
- \(B(1, 1)\)

### Step 4: Check which vertex lies on the line.
Substituting the coordinates of vertex \(A(-3, 1)\) into the line equation:
\[ 4(-3) + 7(1) + 5 = -12 + 7 + 5 = 0 \]
This means point \(A\) lies on the line.

Now, check vertex \(B(1, 1)\):
\[ 4(1) + 7(1) + 5 = 4 + 7 + 5 = 16 \neq 0 \]
So, point \(B\) does not lie on the line.

### Step 5: Identify the other vertices of the rectangle.
Since \(AB\) is horizontal (both points have the same y-coordinate), the other two vertices \(C\) and \(D\) will have the same y-coordinate as \(A\) and \(B\) respectively, and will lie on the line perpendicular to \(AB\) (which is vertical).

### Step 6: Find the slope of the perpendicular lines.
The slope of line \(AB\) is \(0\) (horizontal), thus the slopes of lines \(BC\) and \(AD\) must be undefined (vertical lines).

### Step 7: Find the equations of the other sides of the rectangle.
1. **Line \(CD\)**: This line is parallel to \(AB\) and has the same slope as the line \(4x + 7y + 5 = 0\). The equation for line \(CD\) can be derived from the point-slope form using point \(B(1, 1)\):
\[ 4x + 7y = k \quad (k \text{ is determined by point } B) \]
Substituting \(B(1, 1)\):
\[ 4(1) + 7(1) = k \implies k = 11 \]
Thus, the equation of line \(CD\) is:
\[ 4x + 7y = 11 \]

2. **Line \(BC\)**: The slope of line \(BC\) is \(7/4\) (perpendicular to \(AB\)). Using point \(B(1, 1)\):
\[ y - 1 = \frac{7}{4}(x - 1) \]
Rearranging gives:
\[ 7x - 4y = 3 \]

3. **Line \(AD\)**: Similarly, using point \(A(-3, 1)\) and the slope \(7/4\):
\[ y - 1 = \frac{7}{4}(x + 3) \]
Rearranging gives:
\[ 7x - 4y = -25 \]

### Step 8: Compare with the options.
Now we have the equations:
1. \(4x + 7y = 11\) (line \(CD\))
2. \(7x - 4y = 3\) (line \(BC\))
3. \(7x - 4y = -25\) (line \(AD\))

We need to identify which of the given options is not one of these equations.

### Conclusion
After checking the options, we find that the equation that does not match any of the derived equations is the answer.