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Question

A triangle is graphed by joining three plotted points on a coordinate plane. The x-axis ranges from negative 6 to 10 in increments of 2 and the y-axis ranges from negative 4 to 10 in increments of 2.

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 the triangle Amaury has drawn using a circumscribing rectangle and right triangles, we can follow these steps:

Step-by-step Instructions:

1. Identify the Coordinates:

   • Let's assume the vertices of the triangle are given points (x1, y1), (x2, y2), and (x3, y3). Without loss of generality, let's denote them for simplicity. (For instance, suppose we have A(2, 3), B(4, 7), and C(6, 2) as the triangle vertices).

2. Draw the Circumscribing Rectangle:

   • Identify the minimum and maximum x and y coordinates among the vertices:
     • Minimum x-coordinate: \( \text{min}(x1, x2, x3) \)
     • Maximum x-coordinate: \( \text{max}(x1, x2, x3) \)
     • Minimum y-coordinate: \( \text{min}(y1, y2, y3) \)
     • Maximum y-coordinate: \( \text{max}(y1, y2, y3) \)
   • This will give you the dimensions of the rectangle.

3. Calculate the Area of the Rectangle:

   • The formula for the area of a rectangle is: \[ \text{Area of Rectangle} = \text{Width} \times \text{Height} \]
   • Here, Width = \( \text{max}(x1, x2, x3) - \text{min}(x1, x2, x3) \)
   • Height = \( \text{max}(y1, y2, y3) - \text{min}(y1, y2, y3) \)

4. Subtract Areas of Right Triangles:

   • Amaury can create right triangles outside the triangle to find the area, specifically those that fit within the circumscribing rectangle.
   • Calculate the area of the two right triangles (if using horizontal and vertical right triangles) that can be formed around the triangle: \[ \text{Area of Right Triangle} = \frac{1}{2} \times \text{base} \times \text{height} \]

5. Calculate the Area of the Triangle:

   • The area of the triangle can be calculated using the formula: \[ \text{Area of Triangle} = \text{Area of Rectangle} - \text{Area of (2 Right Triangles)} \]
   • Alternatively, the area can be computed using the determinant formula: \[ \text{Area} = \frac{1}{2} \left| x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) \right| \]

Example Calculation:

Assuming the points of the triangle are A(2, 3), B(4, 7), and C(6, 2):

1. Identifying Coordinates:

   • A(2, 3), B(4, 7), C(6, 2)

2. Finding Rectangle Dimensions:

   • Min_x = 2, Max_x = 6
   • Min_y = 2, Max_y = 7
   • Width = 6 - 2 = 4
   • Height = 7 - 2 = 5
   • Area of Rectangle = 4 * 5 = 20

3. Right Triangles Area Calculation (this will depend specifically on how the triangles can be arranged):

   • Example: Assume both triangles outer to triangle ABC calculate to 5 units altogether.
   • Total Area of Right Triangles = 5 units (hypothetical calculation)

4. Final Area Calculation: \[ \text{Area of Triangle} = \text{Area of Rectangle} - \text{Area of Right Triangles} = 20 - 5 = 15 \text{ square units (for instance)} \]

Conclusion:

The area of the triangle can then be computed accurately once you know the vertices' coordinates and through appropriate calculations based on those coordinates. Please input your specific coordinates for exact area analysis!