What is the area of the triangle formed by points (0,0), (3,4), (4,3)?

Options

A. `4 " unit"^2`
B. `7//2 " unit"^2`
C. `5//2" unit"^2`
D. `5//3" unit"^2`

Answer

To find the area of the triangle formed by the points (0,0), (3,4), and (4,3), we can use the formula for the area of a triangle given 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: Identify the coordinates
Let the points be:
- \( A(0, 0) \) → \( (x_1, y_1) = (0, 0) \)
- \( B(3, 4) \) → \( (x_2, y_2) = (3, 4) \)
- \( C(4, 3) \) → \( (x_3, y_3) = (4, 3) \)

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

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

### Step 3: Simplify the expression
Calculating each term:
- First term: \( 0(4 - 3) = 0 \)
- Second term: \( 3(3 - 0) = 9 \)
- Third term: \( 4(0 - 4) = -16 \)

Now, substituting these values back into the area formula:

\[ \text{Area} = \frac{1}{2} \left| 0 + 9 - 16 \right| \]

### Step 4: Calculate the absolute value
Calculating the expression inside the absolute value:

\[ 0 + 9 - 16 = -7 \]

Taking the absolute value:

\[ \left| -7 \right| = 7 \]

### Step 5: Final calculation of the area
Now, we can calculate the area:

\[ \text{Area} = \frac{1}{2} \times 7 = \frac{7}{2} \]

### Conclusion
Thus, the area of the triangle formed by the points (0,0), (3,4), and (4,3) is:

\[ \text{Area} = \frac{7}{2} \text{ square units} \]

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