The line `3x+2y=24` meets the y-axis at `A` and the x-axis at `Bdot` The perpendicular bisector of `A B` meets the line through `(0,-1)` parallel to the x-axis at `Cdot` If the area of triangle `A B C` is `A` , then the value of `A/(13)` is________

Explanation

To solve the problem step by step, we will follow the instructions given in the video transcript and derive the required values systematically.

### Step 1: Find the Points A and B
The line given is \(3x + 2y = 24\).

**Finding Point A (y-intercept):**
To find where the line meets the y-axis, set \(x = 0\):
\[ 3(0) + 2y = 24 \implies 2y = 24 \implies y = 12 \]
Thus, point \(A\) is \((0, 12)\).

**Finding Point B (x-intercept):**
To find where the line meets the x-axis, set \(y = 0\):
\[ 3x + 2(0) = 24 \implies 3x = 24 \implies x = 8 \]
Thus, point \(B\) is \((8, 0)\).

### Step 2: Find the Midpoint of AB
The midpoint \(M\) of segment \(AB\) can be calculated using the midpoint formula:
\[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) = \left(\frac{0 + 8}{2}, \frac{12 + 0}{2}\right) = (4, 6) \]

### Step 3: Find the Slope of AB
The slope \(m\) of line segment \(AB\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 12}{8 - 0} = \frac{-12}{8} = -\frac{3}{2} \]

### Step 4: Find the Slope of the Perpendicular Bisector
The slope of the perpendicular bisector is the negative reciprocal of the slope of \(AB\):
\[ m_{perpendicular} = -\frac{1}{m} = \frac{2}{3} \]

### Step 5: Write the Equation of the Perpendicular Bisector
Using point-slope form \(y - y_1 = m(x - x_1)\) with point \(M(4, 6)\):
\[ y - 6 = \frac{2}{3}(x - 4) \]
Expanding this:
\[ y - 6 = \frac{2}{3}x - \frac{8}{3} \implies y = \frac{2}{3}x + \frac{10}{3} \]

### Step 6: Find Point C
Point \(C\) lies on the line \(y = -1\) (a horizontal line through \((0, -1)\)). Set \(y = 0\) in the equation of the perpendicular bisector:
\[ 0 = \frac{2}{3}x + \frac{10}{3} \]
Solving for \(x\):
\[ \frac{2}{3}x = -\frac{10}{3} \implies x = -5 \]
Thus, point \(C\) is \((-5, 0)\).

### Step 7: Calculate the Area of Triangle ABC
The area \(A\) of triangle \(ABC\) can be calculated using the formula:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting the coordinates \(A(0, 12)\), \(B(8, 0)\), and \(C(-5, 0)\):
\[ A = \frac{1}{2} \left| 0(0 - 0) + 8(0 - 12) + (-5)(12 - 0) \right| \]
\[ = \frac{1}{2} \left| 0 - 96 - 60 \right| = \frac{1}{2} \left| -156 \right| = \frac{156}{2} = 78 \]

### Step 8: Find the Value of \(\frac{A}{13}\)
Now, we need to find:
\[ \frac{A}{13} = \frac{78}{13} = 6 \]

### Final Answer
Thus, the value of \(\frac{A}{13}\) is \(6\).

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