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