Question

Use the image to answer the question. Amaury draws an acute triangle on the coordinate grid. He finds the area of the triangle by drawing a rectangle that circumscribes the triangle and forming right triangles. How can Amaury find the area of the triangle? What is the area of the triangle in square units? A(-3,5) B (3,7) C(2,3) (4 points)

Answer 1

To find the area of the triangle formed by points A(-3, 5), B(3, 7), and C(2, 3) using the coordinates, you can use the formula for the area of a triangle given its vertices:

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

Where \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) are the coordinates of points A, B, and C.

Using the points:
- A = (-3, 5)
- B = (3, 7)
- C = (2, 3)

Substituting into the formula:

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

Calculating each term:

1. \( -3(7 - 3) = -3 \times 4 = -12 \)
2. \( 3(3 - 5) = 3 \times (-2) = -6 \)
3. \( 2(5 - 7) = 2 \times (-2) = -4 \)

Putting it all together:

\[
\text{Area} = \frac{1}{2} \left| -12 - 6 - 4 \right| = \frac{1}{2} \left| -22 \right| = \frac{1}{2} \times 22 = 11
\]

Therefore, the area of the triangle is \( 11 \) square units.