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.
-
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 \( A_{ADB} = \frac{1}{2} \times b \times h = \frac{1}{2} \cdot 2 \cdot 0 = 0 \).
-
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 \( A_{BEC} = \frac{1}{2} \times 4 \times 1 = 2 \).
-
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: \[ \boxed{17} \]