Find the area of that triangle whose vertices are `(2,3),(-3,4)and(7,5).`

Answer

To find the area of the triangle with vertices at the points \( (2,3) \), \( (-3,4) \), and \( (7,5) \), we can use the formula for the area of a triangle given by the coordinates of its vertices:

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

### Step 1: Assign the coordinates
Let:
- \( (x_1, y_1) = (2, 3) \)
- \( (x_2, y_2) = (-3, 4) \)
- \( (x_3, y_3) = (7, 5) \)

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

\[ \text{Area} = \frac{1}{2} \left| 2(4 - 5) + (-3)(5 - 3) + 7(3 - 4) \right| \]

### Step 3: Calculate each term
Now, calculate each term:
- First term: \( 2(4 - 5) = 2 \times (-1) = -2 \)
- Second term: \( -3(5 - 3) = -3 \times 2 = -6 \)
- Third term: \( 7(3 - 4) = 7 \times (-1) = -7 \)

### Step 4: Combine the terms
Now combine the terms:

\[ \text{Area} = \frac{1}{2} \left| -2 - 6 - 7 \right| = \frac{1}{2} \left| -15 \right| = \frac{1}{2} \times 15 = 7.5 \]

### Step 5: Final area
Thus, the area of the triangle is:

\[ \text{Area} = 7.5 \text{ square units} \]