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)

Answer 1

To find the area of triangle \( ABC \) with vertices \( A(-3, 5) \), \( B(3, 7) \), and \( C(2, 3) \), we can follow the steps specified in your request.

Step 1: Determine the dimensions of the circumscribing rectangle

First, we need to determine the bounds for our rectangle based on the coordinates of the triangle's vertices.

• The minimum and maximum x-coordinates are:

  • \( \text{min } x = -3 \) (from point A)
  • \( \text{max } x = 3 \) (from point B)

• The minimum and maximum y-coordinates are:

  • \( \text{min } y = 3 \) (from point C)
  • \( \text{max } y = 7 \) (from point B)

Thus, the rectangle will have the following corners:

• Bottom-left: \( (-3, 3) \)
• Bottom-right: \( (3, 3) \)
• Top-left: \( (-3, 7) \)
• Top-right: \( (3, 7) \)

Step 2: Calculate the area of the rectangle

The width of the rectangle can be calculated as: \[ \text{Width} = \text{max } x - \text{min } x = 3 - (-3) = 6 \]

The height of the rectangle is: \[ \text{Height} = \text{max } y - \text{min } y = 7 - 3 = 4 \]

Thus, the area of the rectangle \( A_R \) is: \[ A_R = \text{Width} \times \text{Height} = 6 \times 4 = 24 \]

Step 3: Calculate the area of each right triangle formed between \( \triangle ABC \) and the rectangle

We can identify the right triangles formed at each vertex of the triangle and the rectangle.

1. Right Triangle at vertex A:

   • Base = height of rectangle - y-coordinate of A = \( 7 - 5 = 2 \)
   • Height = x-coordinate of A - min x = \( -3 - (-3) = 0 \) (the height at y = 3)

   Area of triangle at A: \[ A_A = \frac{1}{2} \times 2 \times 0 = 0 \]

2. Right Triangle at vertex B:

   • Base = y-coordinate of B - height of rectangle = \( 7 - 7 = 0 \)
   • Height = max x - x-coordinate of B = \( 3 - 3 = 0 \)

   Area of triangle at B: \[ A_B = \frac{1}{2} \times 0 \times 0 = 0 \]

3. Right Triangle at vertex C:

   • Base = height of rectangle - y-coordinate of C = \( 7 - 3 = 4 \)
   • Height = x-coordinate of C - min x = \( 2 - (-3) = 5 \)

   Area of triangle at C: \[ A_C = \frac{1}{2} \times 5 \times 4 = 10 \]

Step 4: Calculate the area of triangle \( ABC \)

The area of triangle \( ABC \), denoted \( A_{ABC} \), can be found using the formula: \[ A_{ABC} = A_R - (A_A + A_B + A_C) \] Substituting the values we calculated earlier: \[ A_{ABC} = 24 - (0 + 0 + 10) = 24 - 10 = 14 \]

Conclusion

Therefore, the area of triangle \( ABC \) is: \[ \boxed{14} \]