Find the area of △ABC by drawing a rectangle that circumscribes the triangle, forming right triangles between △ABC and the rectangle. Calculate the area of the rectangle. Calculate the area of each right triangle formed between △ABC and the rectangle. Use this to then find the area of △ABC . (A -3,5 B 3,7 C 2,3) simplest way

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To find the area of triangle \( \triangle ABC \) with vertices \( A(-3, 5) \), \( B(3, 7) \), and \( C(2, 3) \) using a circumscribing rectangle, we can follow these steps:

Step 1: Identify the Rectangle

The rectangle must circumscribe the triangle with vertices at the extremes of the triangle's coordinates.

  • The minimum x-coordinate is \( -3 \) (from point \( A \)).
  • The maximum x-coordinate is \( 3 \) (from point \( B \)).
  • The minimum y-coordinate is \( 3 \) (from point \( C \)).
  • The maximum y-coordinate is \( 7 \) (from point \( B \)).

Step 2: Calculate the Dimensions of the Rectangle

Using the extreme coordinates derived:

  • Width (horizontal length) of the rectangle = \( 3 - (-3) = 6 \)
  • Height (vertical length) of the rectangle = \( 7 - 3 = 4 \)

Step 3: Calculate the Area of the Rectangle

The area \( A_{rectangle} \) of the rectangle is given by the formula: \[ A_{rectangle} = \text{width} \times \text{height} = 6 \times 4 = 24 \]

Step 4: Calculate the Area of Triangles Outside Triangle ABC

Next, we calculate the area of the three right triangles formed between the triangle \( \triangle ABC \) and the rectangle.

  1. Triangle ADB where \( D \) is at the point \( (-3, 3) \).

    • Base = \( 7 - 5 = 2 \)
    • Height = \( 3 - (-3) = 0 \) (The triangle ADB has a height measurement on the y-axis)

    Thus, the area \( A_{ADB} = \frac{1}{2} \times b \times h = \frac{1}{2} \cdot 2 \cdot 0 = 0 \).

  2. Triangle BEC where \( E \) is at the point \( (3, 3) \).

    • Base = \( 7 - 3 = 4 \)
    • Height = \( 2 - 3 = -1 \) (But measuring height in absolute value \( = 1 \))

    Thus, the area \( A_{BEC} = \frac{1}{2} \times 4 \times 1 = 2 \).

  3. Triangle CFD where \( F \) is at the point \( (2, 3) \).

    • Base = \( 2 - (-3) = 5 \)
    • Height = \( 5 - 3 = 2 \)

    Thus, the area \( A_{CDF} = \frac{1}{2} \times 5 \times 2 = 5 \).

Step 5: Calculate Area of Triangle ABC

The remaining area of the rectangle is the area of triangle \( \triangle ABC \).

\[ A_{ABC} = A_{rectangle} - (A_{ADB} + A_{BEC} + A_{CDF}) \]

Summing the areas of the right triangles: \[ A_{right} = 0 + 2 + 5 = 7 \]

Thus, the area of triangle \( ABC \) is: \[ A_{ABC} = 24 - 7 = 17 \]

The final area of triangle \( ABC \) is: \[ \boxed{17} \]