Question

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 \( \triangle ABC \) using the rectangle that circumscribes it, we will follow these steps:

1. **Find the coordinates of points A, B, and C**:
- \( A(-3, 5) \)
- \( B(3, 7) \)
- \( C(2, 3) \)

2. **Determine the coordinates of the vertices of the rectangle**:
The rectangle will be defined by the minimum and maximum \(x\) and \(y\) values of points A, B, and C.

- Minimum \(x\) value: \( -3 \) (A)
- Maximum \(x\) value: \( 3 \) (B)
- Minimum \(y\) value: \( 3 \) (C)
- Maximum \(y\) value: \( 7 \) (B)

Thus, the rectangle has vertices at:
- \( (-3, 3) \) (bottom-left)
- \( (3, 3) \) (bottom-right)
- \( (3, 7) \) (top-right)
- \( (-3, 7) \) (top-left)

3. **Calculate the dimensions of the rectangle**:
- Width (horizontal side): \( 3 - (-3) = 6 \)
- Height (vertical side): \( 7 - 3 = 4 \)

4. **Calculate the area of the rectangle**:
\[
\text{Area of the rectangle} = \text{width} \times \text{height} = 6 \times 4 = 24
\]

5. **Calculate the areas of each right triangle formed between \( \triangle ABC \) and the rectangle**:
We will determine the area of each triangle formed by the vertices of \( \triangle ABC \) and the corners of the rectangle.

- Right triangle formed with corner at \( (-3, 3) \):
- Base = \( |-3 - 2| = 1 \) (horizontal distance from C to left side)
- Height = \( |7 - 5| = 2 \) (vertical distance from A to top side)

\[
\text{Area of triangle 1} = \frac{1}{2} \times 1 \times 2 = 1
\]

- Right triangle formed with corner at \( (3, 3) \):
- Base = \( |3 - 2| = 1 \)
- Height = \( |7 - 3| = 4 \)

\[
\text{Area of triangle 2} = \frac{1}{2} \times 1 \times 4 = 2
\]

- Right triangle formed with corner at \( (3, 7) \):
- Base = \( |3 - 3| = 0 \) (No area because the base is zero)
- Height = \( |7 - 3| = 4 \)

\[
\text{Area of triangle 3} = 0
\]

- Right triangle formed with corner at \( (-3, 7) \):
- Base = \( |-3 - (-3)| = 0 \) (No area because the base is zero)
- Height = \( |7 - 3| = 4 \)

\[
\text{Area of triangle 4} = 0
\]

6. **Sum the areas of the right triangles and subtract from the area of the rectangle**:
The total area of the right triangles is:
\[
\text{Total area of triangles} = 1 + 2 + 0 + 0 = 3
\]

7. **Find the area of triangle \( \triangle ABC \)**:
\[
\text{Area of } \triangle ABC = \text{Area of rectangle} - \text{Total area of triangles}
\]
\[
\text{Area of } \triangle ABC = 24 - 3 = 21
\]

Therefore, the area of triangle \( \triangle ABC \) is \( \boxed{21} \).