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 $\mathrm{△}ABC$ 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 $\mathrm{△}ABC$ and the rectangle: We will determine the area of each triangle formed by the vertices of $\mathrm{△}ABC$ 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 $\mathrm{△}ABC$: $$\text{Area of }\mathrm{△}ABC=\text{Area of rectangle}-\text{Total area of triangles}$$ $$\text{Area of }\mathrm{△}ABC=24-3=21$$

Therefore, the area of triangle $\mathrm{△}ABC$ is $\boxed{{\displaystyle 21}}$.