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

Let's solve the problem with the updated coordinates. The vertices of the triangle are \((-2, 0)\), \((4, 0)\), and \((3, y)\). The area of the triangle is given as 9.

We can use the formula for the area of a triangle:

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

Substituting the given coordinates into the formula:

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

Simplifying the expression:

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

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

\[
\text{Area} = \frac{1}{2} \left| 6y \right|
\]

\[
\text{Area} = \frac{6}{2} \times |y| = 3|y|
\]

Given that the area is 9:

\[
3|y| = 9
\]

\[
|y| = \frac{9}{3} = 3
\]

This means \( y \) can be either 3 or -3.

Thus, the possible values for \( y \) are \( y = 3 \) or \( y = -3 \).

Would you like further clarification on any of these steps?

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Here are five related questions for you:

1. How would the area formula change if the vertices were in three dimensions?
2. What is the geometric interpretation of the absolute value in this context?
3. How can you determine the height of the triangle from its area and base?
4. What are other methods to find the area of a triangle given its vertices?
5. How does shifting one of the vertices along the x-axis affect the area?

**Tip:** When dealing with absolute values in geometry, always check for both positive and negative possibilities to cover all solutions.

Discover how to determine the y-coordinate of a triangle with vertices at (-2, 0), (4, 0), and (3, y) given its area of 9. Follow step-by-step calculations using the formula for the area of a triangle with vertices. Suitable for Grades 9-12.

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