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B (3, 7)

A (-3, 5)
-10
-8
-6
-4
-2
-2-
-0-
-2-
C(2, 3)
2
4
8
10
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?
(4 points)
Ignore the extra numbers

3 answers

To find the area of the acute triangle formed by the points A(-3, 5), B(3, 7), and C(2, 3), we can use the formula for the area of a triangle given by three vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):

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

Substituting the coordinates of the points A, B, and C into the formula:

  • \(A(-3, 5)\): \(x_1 = -3\), \(y_1 = 5\)
  • \(B(3, 7)\): \(x_2 = 3\), \(y_2 = 7\)
  • \(C(2, 3)\): \(x_3 = 2\), \(y_3 = 3\)

Now, we can plug in the values:

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

Calculating each term step-by-step:

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

So we combine these results:

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

The area of the triangle is \( \boxed{11} \) square units.

Remove all the extra lines and stuff

To find the area of the triangle formed by the points A(-3, 5), B(3, 7), and C(2, 3), we use the formula:

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

Substituting the coordinates:

  • For A(-3, 5): \(x_1 = -3\), \(y_1 = 5\)
  • For B(3, 7): \(x_2 = 3\), \(y_2 = 7\)
  • For C(2, 3): \(x_3 = 2\), \(y_3 = 3\)

The area calculation is:

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

Calculating each term:

\(-3(4) + 3(-2) + 2(-2) = -12 - 6 - 4 = -22\)

Thus:

\[ \text{Area} = \frac{1}{2} \left| -22 \right| = \frac{1}{2} \cdot 22 = 11 \]

The area of the triangle is \( \boxed{11} \) square units.