To find the area of triangle ABC with vertices A(-3, 5), B(3, 7), and C(2, 3) using a rectangle that circumscribes the triangle, we can follow these steps:
Step 1: Determine the coordinates and the rectangle
The coordinates of the vertices are:
- A(-3, 5)
- B(3, 7)
- C(2, 3)
Now, let's find the extreme coordinates to define the rectangle:
- 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).
Thus the rectangle will have the following corners:
- Bottom-left corner: (-3, 3)
- Top-right corner: (3, 7)
Step 2: Calculate the area of the rectangle
The width of the rectangle (difference in x-coordinates):
\[
\text{Width} = 3 - (-3) = 6
\]
The height of the rectangle (difference in y-coordinates):
\[
\text{Height} = 7 - 3 = 4
\]
The area of the rectangle is:
\[
\text{Area of rectangle} = \text{Width} \times \text{Height} = 6 \times 4 = 24
\]
Step 3: Find the areas of the right triangles
Next, we will determine the area of the right triangles formed between the triangle ABC and the rectangle. Note that:
- Each vertex of the triangle connects to the corresponding edges of the rectangle, forming three right triangles.
Triangle at vertex A(-3, 5):
- Right triangle with base along line y = 3 and height along line x = -3.
- Base length (y difference): 5 - 3 = 2
- Height length (x difference): 0 - (-3) = 0 (no width).
Area of triangle at A:
\[
\text{Area}_A = \frac{1}{2} \times 3 \times 2 = 3
\]
Triangle at vertex B(3, 7):
- Right triangle with base along line y = 7 and height along line x = 3.
- Base length (y difference): 7 - 7 = 0 (no height).
- Height length (x difference): 3 - 0 = 3.
Area of triangle at B:
\[
\text{Area}_B = \frac{1}{2} \times 3 \times 0 = 0
\]
Triangle at vertex C(2, 3):
- Right triangle with base along line y = 3 and height along line x = 2.
- Base length (y difference): 3 - 3 = 0 (no height).
- Height length (x difference): 2 - 0 = 2.
Area of triangle at C:
\[
\text{Area}_C = \frac{1}{2} \times 2 \times 0 = 0
\]
Step 4: Calculate the area of triangle ABC
Now, the area of triangle ABC can be found by subtracting the areas of the right triangles formed from the area of the rectangle.
Let \( \text{Area of triangle ABC} = \text{Area of rectangle} - (\text{Area}_A + \text{Area}_B + \text{Area}C) \): \[ \text{Area}{ABC} = 24 - (3 + 0 + 0) = 21 \]
Final Result
Thus, the area of triangle ABC is: \[ \text{Area}_{ABC} = 21 \]
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