To solve the problem, we need to find the area of triangle ABC where point A is given and points B and C lie on a specific line. Here are the steps to find the solution:
### Step 1: Identify the line equation
The line is given by the equation \((x + 2)/3 = (y - 1)/0 = z/4\). This means that:
- \(y = 1\) (since the denominator for \(y\) is 0, \(y\) is constant)
- \(x + 2 = 3\lambda\) (where \(\lambda\) is a parameter)
- \(z = 4\lambda\)
From this, we can express the coordinates of any point on the line as:
- \(x = 3\lambda - 2\)
- \(y = 1\)
- \(z = 4\lambda\)
### Step 2: Define points B and C
Let:
- Point B: \(B(3\lambda_1 - 2, 1, 4\lambda_1)\)
- Point C: \(C(3\lambda_2 - 2, 1, 4\lambda_2)\)
### Step 3: Calculate the distance BC
The distance \(BC\) is given to be 5 units. We can calculate this distance using the distance formula:
\[
BC = \sqrt{(x_C - x_B)^2 + (y_C - y_B)^2 + (z_C - z_B)^2}
\]
Substituting the coordinates of B and C:
\[
BC = \sqrt{((3\lambda_2 - 2) - (3\lambda_1 - 2))^2 + (1 - 1)^2 + (4\lambda_2 - 4\lambda_1)^2}
\]
This simplifies to:
\[
BC = \sqrt{(3(\lambda_2 - \lambda_1))^2 + (4(\lambda_2 - \lambda_1))^2}
\]
\[
= \sqrt{9(\lambda_2 - \lambda_1)^2 + 16(\lambda_2 - \lambda_1)^2}
\]
\[
= \sqrt{25(\lambda_2 - \lambda_1)^2} = 5|\lambda_2 - \lambda_1|
\]
Setting this equal to 5 gives us:
\[
5|\lambda_2 - \lambda_1| = 5 \implies |\lambda_2 - \lambda_1| = 1
\]
Thus, we can set \(\lambda_2 = \lambda_1 + 1\) or \(\lambda_2 = \lambda_1 - 1\).
### Step 4: Find the coordinates of B and C
Assuming \(\lambda_1 = \lambda\), we have:
- \(B(3\lambda - 2, 1, 4\lambda)\)
- \(C(3(\lambda + 1) - 2, 1, 4(\lambda + 1)) = (3\lambda + 1, 1, 4\lambda + 4)\)
### Step 5: Calculate the area of triangle ABC
The area of triangle ABC can be calculated using the formula:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base \(BC = 5\) and the height \(AD\) can be calculated using the coordinates of point A \(A(1, -1, 2)\) and the line containing points B and C.
### Step 6: Find the height AD
To find the height \(AD\), we need to find the distance from point A to the line BC. The direction ratios of line BC are \((3, 0, 4)\). The coordinates of point A are \(A(1, -1, 2)\).
Using the formula for the distance from a point to a line in 3D, we can calculate the height \(AD\).
### Step 7: Final calculation
After calculating the height \(AD\), we can substitute it back into the area formula:
\[
\text{Area} = \frac{1}{2} \times 5 \times AD
\]
