The area of triangle whose vertices are ` (a, a), (a + 1, a + 1), (a+2, a)` is :

Options

A. ` a^(3)`
B. `2a`
C. 1
D. 2

Answer

To find the area of the triangle with vertices at the points \( (a, a) \), \( (a + 1, a + 1) \), and \( (a + 2, a) \), we can use the formula for the area of a triangle given by its vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):

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

### Step 1: Identify the coordinates
Let:
- \( (x_1, y_1) = (a, a) \)
- \( (x_2, y_2) = (a + 1, a + 1) \)
- \( (x_3, y_3) = (a + 2, a) \)

### Step 2: Substitute the coordinates into the area formula
Substituting the coordinates into the area formula:

\[ \text{Area} = \frac{1}{2} \left| a((a + 1) - a) + (a + 1)(a - a) + (a + 2)(a - (a + 1)) \right| \]

### Step 3: Simplify the expression
Now, simplify each term inside the absolute value:

1. The first term:
\[ a((a + 1) - a) = a(1) = a \]

2. The second term:
\[ (a + 1)(a - a) = (a + 1)(0) = 0 \]

3. The third term:
\[ (a + 2)(a - (a + 1)) = (a + 2)(-1) = - (a + 2) = -a - 2 \]

Combining these results, we have:
\[ \text{Area} = \frac{1}{2} \left| a + 0 - (a + 2) \right| = \frac{1}{2} \left| a - a - 2 \right| = \frac{1}{2} \left| -2 \right| = \frac{1}{2} \times 2 = 1 \]

### Final Result
Thus, the area of the triangle is:
\[ \text{Area} = 1 \]