Question

To find the area of the triangle formed by the points \((-3, 5)\), \((3, 7)\), and \((2, 3)\), we can use the formula for the area of a triangle given by its vertex coordinates:

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

Let's assign the coordinates so that:

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

Now we substitute these values into the formula step-by-step:

1. Calculate \(y_2 - y_3\):

\[
y_2 - y_3 = 7 - 3 = 4
\]

2. Calculate \(y_3 - y_1\):

\[
y_3 - y_1 = 3 - 5 = -2
\]

3. Calculate \(y_1 - y_2\):

\[
y_1 - y_2 = 5 - 7 = -2
\]

Now substitute these values into the area formula:

\[
\text{Area} = \frac{1}{2} \left| (-3)(4) + (3)(-2) + (2)(-2) \right|
\]

Calculating each term:

- The first term: \((-3)(4) = -12\)
- The second term: \((3)(-2) = -6\)
- The third term: \((2)(-2) = -4\)

Now sum these results:

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

Thus, the area of the triangle is:

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

Can you tell me what this is saying

Answer 1

The text describes the process of calculating the area of a triangle given its vertex coordinates in the Cartesian plane. The triangle is formed by the points \((-3, 5)\), \((3, 7)\), and \((2, 3)\).

Here's a breakdown of what the text conveys:

1. **Area Formula**: The area of a triangle can be computed using a specific formula that involves the coordinates of its vertices. The formula provided is:

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

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

2. **Assigning Vertex Coordinates**: The coordinates of the triangle's vertices are assigned to variables \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \).

3. **Calculating Differences in `y` Values**:
- The differences between the `y` coordinates of the vertices are calculated:
- \(y_2 - y_3\): Results in \(4\).
- \(y_3 - y_1\): Results in \(-2\).
- \(y_1 - y_2\): Results in \(-2\).

4. **Substituting into the Area Formula**: These differences are substituted back into the area formula step by step:
- Each term is calculated:
- First term: \((-3)(4) = -12\)
- Second term: \((3)(-2) = -6\)
- Third term: \((2)(-2) = -4\)

5. **Summing the Results**: The total from the substituted terms is summed:
- The expression simplifies to \(-12 - 6 - 4 = -22\).

6. **Calculating the Area**: The absolute value is taken, and the final area is computed:
- The area comes out to be \( \frac{1}{2} \left| -22 \right| = \frac{1}{2} \times 22 = 11\).

7. **Conclusion**: The area of the triangle formed by the three points is \( \boxed{11} \text{ square units} \).

In summary, the text outlines a methodical approach to computing the area of a triangle using its vertex coordinates, demonstrating the calculation clearly for effective understanding.