If the area of triangle with vertices (-3,0), (3,0) and (0,k) is 9 sq . units then the value of k is

Options

A. 9
B. 6
C. 3
D. `-9`

Answer

To find the value of \( k \) for the triangle with vertices at \((-3,0)\), \((3,0)\), and \((0,k)\) such that its area is \( 9 \) square units, we can use the formula for the area of a triangle given its vertices. The area \( A \) of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is given by:

\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

### Step 1: Assign the vertices
Let:
- \( (x_1, y_1) = (-3, 0) \)
- \( (x_2, y_2) = (3, 0) \)
- \( (x_3, y_3) = (0, k) \)

### Step 2: Substitute the coordinates into the area formula
Substituting the coordinates into the area formula, we have:

\[ A = \frac{1}{2} \left| -3(0 - k) + 3(k - 0) + 0(0 - 0) \right| \]

### Step 3: Simplify the expression
Now simplify the expression:

\[ A = \frac{1}{2} \left| -3(-k) + 3k \right| \]
\[ = \frac{1}{2} \left| 3k + 3k \right| \]
\[ = \frac{1}{2} \left| 6k \right| \]
\[ = 3|k| \]

### Step 4: Set the area equal to 9
We know the area is \( 9 \) square units, so we set up the equation:

\[ 3|k| = 9 \]

### Step 5: Solve for \( |k| \)
Dividing both sides by \( 3 \):

\[ |k| = 3 \]

### Step 6: Determine the values of \( k \)
This gives us two possible solutions for \( k \):

\[ k = 3 \quad \text{or} \quad k = -3 \]

### Conclusion
Thus, the values of \( k \) that satisfy the condition are \( k = 3 \) or \( k = -3 \).