Sure, I can help you find the area of the triangle with vertices at \(\((1, 1)\)\), \(\((5, 3)\)\), and \(\((3, 5)\)\). We will use the formula for the area of a triangle when the coordinates of its vertices are known.
1. Label the coordinates of the vertices:
\(\[
(x_1, y_1) = (1, 1), \quad (x_2, y_2) = (5, 3), \quad (x_3, y_3) = (3, 5)
\]\)
2. The formula to find the area of a triangle with vertices at \(\((x_1, y_1)\)\), \(\((x_2, y_2)\)\), and \(\((x_3, y_3)\)\) is:
\(\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]\)
3. Substitute the coordinates into the formula:
\(\[
\text{Area} = \frac{1}{2} \left| 1(3 - 5) + 5(5 - 1) + 3(1 - 3) \right|
\]\)
4. Simplify the expressions inside the absolute value:
\(\[
\text{Area} = \frac{1}{2} \left| 1(-2) + 5(4) + 3(-2) \right|
\]\)
5. Calculate each term separately:
\(\[
\text{Area} = \frac{1}{2} \left| -2 + 20 - 6 \right|
\]\)
6. Combine the terms inside the absolute value:
\(\[
\text{Area} = \frac{1}{2} \left| 12 \right|
\]\)
7. Apply the absolute value (which remains 12 as it's already positive):
\(\[
\text{Area} = \frac{1}{2} \times 12
\]\)
8. Finally, calculate the area:
\(\[
\text{Area} = 6.0
\]\)
So, the area of the triangle with vertices at \(\((1, 1)\)\), \(\((5, 3)\)\), and \(\((3, 5)\)\) is \(\(6.0\)\) square units.
