The area of a triangle whose vertices are (-2, 0), (-4, 0), and (3, y) is 9. Find y.

To find the value of \( y \), we'll use the formula for the area of a triangle given its vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\). The formula is:

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

Given the vertices are \((-2, 0)\), \((-4, 0)\), and \((3, y)\), and the area is 9, we substitute the coordinates into the formula:

\[
\text{Area} = \frac{1}{2} \left| -2(0 - y) + (-4)(y - 0) + 3(0 - 0) \right|
\]

Simplifying this:

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

\[
\text{Area} = \frac{1}{2} \left| 2y - 4y \right|
\]

\[
\text{Area} = \frac{1}{2} \left| -2y \right|
\]

\[
\text{Area} = \frac{1}{2} \times 2 \times |y| = |y|
\]

Given that the area is 9:

\[
|y| = 9
\]

This means \( y \) can be either 9 or -9.

So, \( y = 9 \) or \( y = -9 \).

Would you like further details on any of these steps?

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Here are five related questions for you:

1. How do you find the area of a triangle using the determinant method?
2. What are the conditions under which the area of a triangle can be zero?
3. How does the sign of the area affect the orientation of a triangle's vertices?
4. Can you explain how to calculate the area of a triangle using Heron's formula?
5. How does changing one vertex affect the area of a triangle in the coordinate plane?

**Tip:** When using absolute values in equations, always consider both the positive and negative possibilities.

Discover how to find the value of y for a triangle with vertices at (-2, 0), (-4, 0), and (3, y) using the area formula for triangles. The solution involves applying the formula for the area of a triangle given its vertices and solving for y. Suitable for high school students studying geometry and coordinate geometry.

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