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. How can Amaury find the area of the triangle? What is the area of the triangle in square units if the triangle coordinates are; (-3, 5), (3, 7), and (2, 3)

Answer 1

To find the area of the triangle with vertices at coordinates \((-3, 5)\), \((3, 7)\), and \((2, 3)\), we can use the following formula for the area of a triangle based on its vertices:

\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

Where \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) are the coordinates of the triangle's vertices.

Let's assign the coordinates to \( (x_1, y_1) = (-3, 5) \), \( (x_2, y_2) = (3, 7) \), and \( (x_3, y_3) = (2, 3) \).

Now we can substitute these values into the area formula:

\[ \text{Area} = \frac{1}{2} \left| (-3)(7 - 3) + 3(3 - 5) + 2(5 - 7) \right| \]

Calculating each term inside the absolute value:

1. \( -3(7 - 3) = -3 \cdot 4 = -12 \)
2. \( 3(3 - 5) = 3 \cdot (-2) = -6 \)
3. \( 2(5 - 7) = 2 \cdot (-2) = -4 \)

Now, substituting these values into the formula gives:

\[ \text{Area} = \frac{1}{2} \left| -12 - 6 - 4 \right| = \frac{1}{2} \left| -22 \right| = \frac{1}{2} \cdot 22 = 11 \]

Thus, the area of the triangle is

\[ \boxed{11} \text{ square units.} \]

Answer 2

I don't think I understand, can you explain this differently please?

Answer 3

Of course! Let’s break it down step by step using a simpler explanation.

Finding the Area of a Triangle Given the Vertices

When you have a triangle defined by three points on a coordinate grid, you can find the area using a specific formula. Here’s how we do it with the coordinates of the triangle you provided:

• Vertices:
  • \( A(-3, 5) \)
  • \( B(3, 7) \)
  • \( C(2, 3) \)

Step 1: Use the Area Formula

The formula to find the area of a triangle given its vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is:

\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

Step 2: Substitute the Coordinates

Using your coordinates:

• Let \( (x_1, y_1) = (-3, 5) \)
• Let \( (x_2, y_2) = (3, 7) \)
• Let \( (x_3, y_3) = (2, 3) \)

Now, plug these values into the formula:

\[ \text{Area}= \frac{1}{2} \left| -3(7 - 3) + 3(3 - 5) + 2(5 - 7) \right| \]

Step 3: Calculate Each Part

Now we will calculate each part step by step:

1. For \( -3(7 - 3) \): \[ 7 - 3 = 4 \quad \Rightarrow \quad -3 \cdot 4 = -12 \]

2. For \( 3(3 - 5) \): \[ 3 - 5 = -2 \quad \Rightarrow \quad 3 \cdot -2 = -6 \]

3. For \( 2(5 - 7) \): \[ 5 - 7 = -2 \quad \Rightarrow \quad 2 \cdot -2 = -4 \]

Step 4: Combine the Results

Now, combine these results in the formula: \[ \text{Area}= \frac{1}{2} \left| -12 - 6 - 4 \right| \]

Calculating inside the absolute value: \[ -12 - 6 - 4 = -22 \]

Taking the absolute value: \[ \left| -22 \right| = 22 \]

Step 5: Final Area Calculation

Finally, divide this by 2 to find the area: \[ \text{Area} = \frac{1}{2} \cdot 22 = 11 \]

Conclusion

Thus, the area of the triangle formed by the points \((-3, 5)\), \((3, 7)\), and \((2, 3)\) is:

\[ \boxed{11} \text{ square units.} \]

If you have any further questions or if any part isn’t clear, feel free to ask!

Answer 4

I think the typing style used is what's confusing me, it might help if instead of saying for example; \text{Area}= \frac{1}{2} \left| -12 - 6 - 4 \right|
\] instead using easier to read symbols.

Answer 5

Certainly! I understand that clearer symbols may help. Let's rephrase the explanation using simpler notation here.

Finding the Area of a Triangle Given the Vertices

Given three points (or vertices) of a triangle on a coordinate grid, we can calculate the area using a formula. You provided the points:

• Point A: \( (-3, 5) \)
• Point B: \( (3, 7) \)
• Point C: \( (2, 3) \)

The formula to calculate the area \( A \) of a triangle defined by these points is:

\[ A = \frac{1}{2} \times \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

Step 1: Substitute the Coordinates

Using the points A, B, and C, we have:

• \( (x_1, y_1) = (-3, 5) \)
• \( (x_2, y_2) = (3, 7) \)
• \( (x_3, y_3) = (2, 3) \)

Now we insert these values into the area formula:

\[ A = \frac{1}{2} \times \left| -3(7 - 3) + 3(3 - 5) + 2(5 - 7) \right| \]

Step 2: Calculate Each Part Inside the Absolute Value

1. Calculate \( -3(7 - 3) \):

   • \( 7 - 3 = 4 \)
   • Multiply: \( -3 \times 4 = -12 \)

2. Calculate \( 3(3 - 5) \):

   • \( 3 - 5 = -2 \)
   • Multiply: \( 3 \times -2 = -6 \)

3. Calculate \( 2(5 - 7) \):

   • \( 5 - 7 = -2 \)
   • Multiply: \( 2 \times -2 = -4 \)

Step 3: Combine the Results

Now, combine all these results:

\[ A = \frac{1}{2} \times \left| -12 - 6 - 4 \right| \]

Calculating inside the absolute value:

\[ -12 - 6 - 4 = -22 \]

Taking the absolute value gives us:

\[ \left| -22 \right| = 22 \]

Step 4: Final Area Calculation

Now, substitute this back into the formula:

\[ A = \frac{1}{2} \times 22 = 11 \]

Conclusion

Thus, the area of the triangle is:

\[ \boxed{11} \text{ square units.} \]

Let me know if this format is easier to read or if there's anything else you'd like to discuss!