Question

What is the area of a triangle with vertices at (−4, 1), ​(−7, 5)​ , and ​(0, 1)​ ?

Enter your answer in the box.

units²

Answer 1

To find the area of a triangle given its vertices, we can use the formula:

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

Where \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) are the coordinates of the vertices.

For the triangle with vertices at \((-4, 1)\), \((-7, 5)\), and \((0, 1)\), we assign:

- \((x_1, y_1) = (-4, 1)\)
- \((x_2, y_2) = (-7, 5)\)
- \((x_3, y_3) = (0, 1)\)

Now, we can substitute these values into the area formula:

\[
\text{Area} = \frac{1}{2} | -4(5 - 1) + (-7)(1 - 1) + 0(1 - 5) |
\]

Calculating each term:

1. \( -4(5 - 1) = -4 \times 4 = -16 \)
2. \( -7(1 - 1) = -7 \times 0 = 0 \)
3. \( 0(1 - 5) = 0 \)

Combining these, we have:

\[
\text{Area} = \frac{1}{2} | -16 + 0 + 0 |
\]
\[
\text{Area} = \frac{1}{2} | -16 |
\]
\[
\text{Area} = \frac{1}{2} \times 16 = 8
\]

Thus, the area of the triangle is:

\[
\boxed{8} \text{ units}^2
\]