To find the area of an acute triangle using a circumscribing rectangle, Amaury can follow these steps:
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Identify the Vertices: Write down the coordinates of the triangle's vertices, which we will call \(A(x_1, y_1)\), \(B(x_2, y_2)\), and \(C(x_3, y_3)\).
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Find the Circumscribing Rectangle: Determine the minimum and maximum x-values and y-values from the vertices:
- Minimum x: \(x_{\text{min}} = \min(x_1, x_2, x_3)\)
- Maximum x: \(x_{\text{max}} = \max(x_1, x_2, x_3)\)
- Minimum y: \(y_{\text{min}} = \min(y_1, y_2, y_3)\)
- Maximum y: \(y_{\text{max}} = \max(y_1, y_2, y_3)\)
The corners of the rectangle are then \((x_{\text{min}}, y_{\text{min}})\), \((x_{\text{max}}, y_{\text{min}})\), \((x_{\text{max}}, y_{\text{max}})\), and \((x_{\text{min}}, y_{\text{max}})\).
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Calculate the Area of the Rectangle: The area of the rectangle is calculated as: \[ \text{Area} = (x_{\text{max}} - x_{\text{min}}) \times (y_{\text{max}} - y_{\text{min}}) \]
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Calculate the Area of Each Right Triangle: To find the area of the triangle more easily, Amaury can calculate the area of the rectangle and subtract the areas of the four right triangles formed at the corners.
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Determine the Areas of the Right Triangles: For each corner of the rectangle, the right triangles can be calculated. For each:
- Triangle at \((x_{\text{min}}, y_{\text{min}})\) is formed by the points \(A\), \(B\), and the right angle.
- Triangle at \((x_{\text{max}}, y_{\text{min}})\):
- Triangle at \((x_{\text{min}}, y_{\text{max}})\):
- Triangle at \((x_{\text{max}}, y_{\text{max}})\):
You will need the coordinates of the points to calculate each area specifically.
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Final Area Calculation: The area of the triangle will be the area of the rectangle minus the areas of the four corners' triangles.
However, without specific coordinates, we cannot derive a numerical area. If you can provide the coordinates of the triangle’s vertices, I can help calculate the exact area.
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