Question

Use the coordinates to find the area of the triangle.
Coordinates: (2,7), (7,5), (2,1)

Answer 1

To find the area of a triangle given its vertices, we can use the formula based on the coordinates of the points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):

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

Using the given coordinates:
\((x_1, y_1) = (2, 7)\),
\((x_2, y_2) = (7, 5)\),
\((x_3, y_3) = (2, 1)\).

Substituting the coordinates into the formula:

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

Calculating each term step-by-step:

1. \(2(5 - 1) = 2 \times 4 = 8\)
2. \(7(1 - 7) = 7 \times (-6) = -42\)
3. \(2(7 - 5) = 2 \times 2 = 4\)

Now, combining these:

\[
\text{Area} = \frac{1}{2} \left| 8 - 42 + 4 \right|
\]

Calculating the inside of the absolute value:

\[
8 - 42 + 4 = -30
\]

Taking the absolute value:

\[
|-30| = 30
\]

Finally, calculating the area:

\[
\text{Area} = \frac{1}{2} \times 30 = 15
\]

Thus, the area of the triangle is

\[
\boxed{15}
\]