Question
Use the image to answer the question Amaurty 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. How can Amaury find the area of the triangle? What is the area of the triangle in square units?
Answers
GPT-4o mini
To find the area of an acute triangle using a circumscribed rectangle, Amaurty can follow these steps:
1. **Determine the Vertices**: Identify the coordinates of the triangle's vertices on the coordinate grid.
2. **Draw a Circumscribing Rectangle**: Draw a rectangle that completely encompasses the triangle. The rectangle's sides will align with the axes of the coordinate grid, with its corners being the extreme points of the triangle.
3. **Calculate the Area of the Rectangle**: Measure the length and width of the rectangle. The area can be found using the formula:
\[
\text{Area}_{\text{rectangle}} = \text{length} \times \text{width}
\]
4. **Identify the Area of Right Triangles**: Form right triangles by drawing lines from the triangle's vertices to the rectangle's edges. The area of these right triangles can be calculated using the formula:
\[
\text{Area}_{\text{right triangle}} = \frac{1}{2} \times \text{base} \times \text{height}
\]
5. **Subtract the Areas of the Right Triangles from the Rectangle**: If there are multiple right triangles formed outside of the triangle, calculate their total area and subtract it from the area of the rectangle to find the area of the triangle:
\[
\text{Area}_{\triangle} = \text{Area}_{\text{rectangle}} - \text{Area}_{\text{right triangles}}
\]
### Finding the Area in Square Units
Without the coordinates or further details of the triangle's dimensions, I can't provide a specific numeric value for the area. If you can supply the vertices of the acute triangle or the dimensions of the rectangle, I can help you calculate the exact area. Alternatively, if you have specific values from your image or problem context, feel free to share them!
1. **Determine the Vertices**: Identify the coordinates of the triangle's vertices on the coordinate grid.
2. **Draw a Circumscribing Rectangle**: Draw a rectangle that completely encompasses the triangle. The rectangle's sides will align with the axes of the coordinate grid, with its corners being the extreme points of the triangle.
3. **Calculate the Area of the Rectangle**: Measure the length and width of the rectangle. The area can be found using the formula:
\[
\text{Area}_{\text{rectangle}} = \text{length} \times \text{width}
\]
4. **Identify the Area of Right Triangles**: Form right triangles by drawing lines from the triangle's vertices to the rectangle's edges. The area of these right triangles can be calculated using the formula:
\[
\text{Area}_{\text{right triangle}} = \frac{1}{2} \times \text{base} \times \text{height}
\]
5. **Subtract the Areas of the Right Triangles from the Rectangle**: If there are multiple right triangles formed outside of the triangle, calculate their total area and subtract it from the area of the rectangle to find the area of the triangle:
\[
\text{Area}_{\triangle} = \text{Area}_{\text{rectangle}} - \text{Area}_{\text{right triangles}}
\]
### Finding the Area in Square Units
Without the coordinates or further details of the triangle's dimensions, I can't provide a specific numeric value for the area. If you can supply the vertices of the acute triangle or the dimensions of the rectangle, I can help you calculate the exact area. Alternatively, if you have specific values from your image or problem context, feel free to share them!