Find the area of the region bounded by the points `(3,0), (4,5) and (5,1).`

Answer

To find the area of the region bounded by the points (3,0), (4,5), and (5,1), we can use the formula for the area of a triangle given its vertices. The formula for the area \( A \) of a triangle with vertices at coordinates \((x_1, y_1)\), \((x_2, y_2)\), 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: Identify the coordinates
We have the points:
- \( A(3, 0) \) → \( (x_1, y_1) \)
- \( B(4, 5) \) → \( (x_2, y_2) \)
- \( C(5, 1) \) → \( (x_3, y_3) \)

### Step 2: Substitute the coordinates into the formula
Now we substitute the coordinates into the area formula:

\[ A = \frac{1}{2} \left| 3(5 - 1) + 4(1 - 0) + 5(0 - 5) \right| \]

### Step 3: Simplify the expression
Calculating each term inside the absolute value:

1. \( 3(5 - 1) = 3 \times 4 = 12 \)
2. \( 4(1 - 0) = 4 \times 1 = 4 \)
3. \( 5(0 - 5) = 5 \times (-5) = -25 \)

Now, combine these results:

\[ A = \frac{1}{2} \left| 12 + 4 - 25 \right| \]

\[ A = \frac{1}{2} \left| 16 - 25 \right| = \frac{1}{2} \left| -9 \right| = \frac{1}{2} \times 9 = \frac{9}{2} \]

### Step 4: Final result
Thus, the area of the triangle formed by the points (3,0), (4,5), and (5,1) is:

\[ \boxed{\frac{9}{2}} \text{ square units} \]