Question

Amaury draws an acute triangle on the coordinate grid. He finds the area of the triangle by drawing a rectangle that circumscribes the triangle and forming right triangles. Show the steps Amaury must take to find the area of the triangle. What is the area of the triangle in square units?

Answer 1

To find the area of an acute triangle using a circumscribing rectangle, Amaury can follow these steps:

### Step 1: Identify the Vertices of the Triangle
Let's assume Amaury's acute triangle has vertices at the points A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃).

### Step 2: Determine the Dimensions of the Circumscribing Rectangle
The rectangle that circumscribes the triangle will be aligned with the coordinate axes. The dimensions of this rectangle can be determined as follows:
- The width (length along the x-axis) of the rectangle is the difference between the maximum and minimum x-coordinates of the triangle's vertices:
\[
\text{Width} = \max(x₁, x₂, x₃) - \min(x₁, x₂, x₃)
\]
- The height (length along the y-axis) of the rectangle is the difference between the maximum and minimum y-coordinates:
\[
\text{Height} = \max(y₁, y₂, y₃) - \min(y₁, y₂, y₃)
\]

### Step 3: Calculate the Area of the Circumscribing Rectangle
The area of the rectangle can then be calculated using the width and height:
\[
\text{Area of Rectangle} = \text{Width} \times \text{Height}
\]

### Step 4: Calculate the Area of the Triangle
To find the area of the triangle, Amaury can either use the standard formula for the area of a triangle given the vertices or apply the method of subtracting the areas of the right triangles formed inside the rectangle.

#### Method 1: Using the Vertex Method
The area of the triangle can be calculated using the formula:
\[
\text{Area of Triangle} = \frac{1}{2} \left| x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂) \right|
\]

#### Method 2: Subtracting Areas of Right Triangles
1. Draw additional lines from the vertices of the triangle to the sides of the rectangle, forming right triangles.
2. Calculate the areas of the right triangles you have formed.
3. Subtract the total area of the right triangles from the area of the circumscribing rectangle to find the area of the triangle.

### Example Calculation
Let's consider a specific example where the vertices of the acute triangle are:
- A(1, 2)
- B(4, 5)
- C(3, 1)

**Step 1: Identify Vertices**
- Vertices are already given.

**Step 2: Circumscribing Rectangle Dimensions**
- Minimum x = 1, Maximum x = 4 → Width = 4 - 1 = 3
- Minimum y = 1, Maximum y = 5 → Height = 5 - 1 = 4

**Step 3: Area of the Rectangle**
- Area = Width × Height = 3 × 4 = 12 square units.

**Step 4: Calculate Area of Triangle**
Using the vertex method:
\[
\text{Area of Triangle} = \frac{1}{2} \left| 1(5 - 1) + 4(1 - 2) + 3(2 - 5) \right|
\]
\[
= \frac{1}{2} \left| 1(4) + 4(-1) + 3(-3) \right|
\]
\[
= \frac{1}{2} \left| 4 - 4 - 9 \right| = \frac{1}{2} \left| -9 \right| = \frac{9}{2} = 4.5 \text{ square units}
\]

### Final Area of the Triangle
Thus, the area of the triangle is **4.5 square units**.