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?(4 points)

Answer 1

To find the area of an acute triangle using the method described by Amaury, follow these steps:

1. **Identify the Vertices of the Triangle:**
Let's denote the vertices of the acute triangle as \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \). Ensure that these points define an acute triangle.

2. **Draw the Circumscribing Rectangle:**
You need to create a rectangle that just fits around the triangle. The corners of the rectangle will have coordinates based on the minimum and maximum values of the triangle's vertices' coordinates:
- The \(x\)-coordinates will be determined by \( \min(x_1, x_2, x_3) \) and \( \max(x_1, x_2, x_3) \).
- The \(y\)-coordinates will be determined by \( \min(y_1, y_2, y_3) \) and \( \max(y_1, y_2, y_3) \).

3. **Find the Area of the Rectangle:**
The formula for the area of the rectangle is:
\[
\text{Area}_{\text{rectangle}} = \text{width} \times \text{height}
\]
where:
- width = \( \max(x_1, x_2, x_3) - \min(x_1, x_2, x_3) \)
- height = \( \max(y_1, y_2, y_3) - \min(y_1, y_2, y_3) \)

4. **Calculate the Area of the Triangle:**
Once the rectangle is drawn, you can find the area of the triangle by using the following formula:
\[
\text{Area}_{\text{triangle}} = \text{Area}_{\text{rectangle}} - \text{Area of right triangles}
\]

If you denote the areas of the right triangles that do not cover the triangle, you can compute the areas based on their base and height, adding them together to subtract from the rectangle's area.

5. **Using the Determinant Formula for a Direct Calculation:**
Alternatively, you can use a determinant formula to find the area of the triangle directly without drawing the rectangle and right triangles. The area of a triangle given by vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is given by:
\[
\text{Area}_{\text{triangle}} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]

6. **Final Calculation:**
Substitute the specific coordinates of your triangle into the area formula to find the area in square units.

Let’s assume a specific example with vertices:
- \( A(1, 3) \)
- \( B(4, 5) \)
- \( C(3, 2) \)

Calculating using the determinant method:
\[
\text{Area}_{\text{triangle}} = \frac{1}{2} \left| 1(5 - 2) + 4(2 - 3) + 3(3 - 5) \right|
\]
Calculating each term:
\[
= \frac{1}{2} \left| 1(3) + 4(-1) + 3(-2) \right| = \frac{1}{2} \left| 3 - 4 - 6 \right| = \frac{1}{2} \left| -7 \right| = \frac{7}{2}
\]
Thus, the area of the triangle equals \( \frac{7}{2} \) or \( 3.5 \) square units.

You can follow this process with the specific coordinates of your triangle to find the exact area.