Find the value of x, if the area of the triangle formed by the points (10,2)(-3,-4) and (x,1) is 5.

Answer

To find the value of \( x \) such that the area of the triangle formed by the points \( (10, 2) \), \( (-3, -4) \), and \( (x, 1) \) is 5, we will use the formula for the area of a triangle given by the coordinates of its vertices.

### Step 1: Write the formula for the area of the triangle
The area \( A \) of a triangle formed by points \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) can be calculated using the formula:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
For our points, we have:
- \( (x_1, y_1) = (10, 2) \)
- \( (x_2, y_2) = (-3, -4) \)
- \( (x_3, y_3) = (x, 1) \)

### Step 2: Substitute the coordinates into the area formula
Substituting the coordinates into the area formula gives:
\[ A = \frac{1}{2} \left| 10(-4 - 1) + (-3)(1 - 2) + x(2 - (-4)) \right| \]
This simplifies to:
\[ A = \frac{1}{2} \left| 10(-5) + (-3)(-1) + x(6) \right| \]
\[ A = \frac{1}{2} \left| -50 + 3 + 6x \right| \]
\[ A = \frac{1}{2} \left| 6x - 47 \right| \]

### Step 3: Set the area equal to 5
We know the area is given as 5, so we set up the equation:
\[ \frac{1}{2} \left| 6x - 47 \right| = 5 \]

### Step 4: Solve for \( |6x - 47| \)
Multiplying both sides by 2:
\[ \left| 6x - 47 \right| = 10 \]

### Step 5: Solve the absolute value equation
This gives us two cases to consider:
1. \( 6x - 47 = 10 \)
2. \( 6x - 47 = -10 \)

#### Case 1: \( 6x - 47 = 10 \)
\[ 6x = 10 + 47 \]
\[ 6x = 57 \]
\[ x = \frac{57}{6} = 9.5 \]

#### Case 2: \( 6x - 47 = -10 \)
\[ 6x = -10 + 47 \]
\[ 6x = 37 \]
\[ x = \frac{37}{6} \approx 6.17 \]

### Step 6: Final values of \( x \)
Thus, the two possible values of \( x \) are:
\[ x = 9.5 \quad \text{and} \quad x = \frac{37}{6} \]