Question

Peeta draws a triangle on the coordinate grid. What is the area of their triangle in square units? (1 point) Responses 47.5 47.5 65 65 40.5 40.5 120. A (-7,3) B (5,8) C (0,-2)

Answer 1

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

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

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

Given points:
- \( A(-7, 3) \)
- \( B(5, 8) \)
- \( C(0, -2) \)

Assign:
- \( x_1 = -7, y_1 = 3 \)
- \( x_2 = 5, y_2 = 8 \)
- \( x_3 = 0, y_3 = -2 \)

Now we can substitute these coordinates into the area formula:

\[
\text{Area} = \frac{1}{2} \left| -7(8 - (-2)) + 5((-2) - 3) + 0(3 - 8) \right|
\]

Calculating each term:

1. \( 8 - (-2) = 8 + 2 = 10 \)
2. \( (-2) - 3 = -5 \)
3. The third term, \( 0(3 - 8) = 0 \)

Now substituting back:

\[
\text{Area} = \frac{1}{2} \left| -7 \cdot 10 + 5 \cdot (-5) + 0 \right|
\]

Calculating:

\[
= \frac{1}{2} \left| -70 - 25 + 0 \right|
\]

\[
= \frac{1}{2} \left| -95 \right|
\]

\[
= \frac{1}{2} \cdot 95 = 47.5
\]

Thus, the area of triangle ABC is \( \boxed{47.5} \) square units.