Triangle DEF is similar to triangle ABC, and both are plotted on a coordinate plane (not shown). The vertices of triange DEF are `D (3,2) ,E (3,-1), and F (-1, -1).` Two of triangle ABC's vettices are quadrant of the coordinate plane as vertex D, what is the y-coordinate of vertex A ?

Answer

To solve the problem, we need to find the y-coordinate of vertex A of triangle ABC, given that triangle DEF is similar to triangle ABC and the coordinates of the vertices of triangle DEF are D(3, 2), E(3, -1), and F(-1, -1).

### Step-by-Step Solution:

1. **Identify the Coordinates of Triangle DEF**:
- The vertices of triangle DEF are given as:
- D(3, 2)
- E(3, -1)
- F(-1, -1)

2. **Determine the Length of Side FE**:
- Calculate the distance between points F and E:
- F(-1, -1) to E(3, -1)
- The length of FE = |x-coordinate of E - x-coordinate of F| = |3 - (-1)| = |3 + 1| = 4 units.

3. **Identify the Coordinates of Triangle ABC**:
- Two vertices of triangle ABC are given as:
- B(-8, -3)
- C(8, -3)
- Since both points B and C share the same y-coordinate (-3), they lie on the same horizontal line.

4. **Determine the Length of Side BC**:
- Calculate the distance between points B and C:
- B(-8, -3) to C(8, -3)
- The length of BC = |x-coordinate of C - x-coordinate of B| = |8 - (-8)| = |8 + 8| = 16 units.

5. **Establish the Ratio of Similarity**:
- Since triangles DEF and ABC are similar, the ratio of corresponding sides must be equal.
- The ratio of sides FE to BC is:
- FE : BC = 4 : 16 = 1 : 4.

6. **Determine the Length of Side DE**:
- Calculate the distance between points D and E:
- D(3, 2) to E(3, -1)
- The length of DE = |y-coordinate of D - y-coordinate of E| = |2 - (-1)| = |2 + 1| = 3 units.

7. **Calculate the Length of Side AC**:
- Using the ratio established, we can find the length of side AC:
- DE : AC = 1 : 4
- If DE = 3, then AC = 3 * 4 = 12 units.

8. **Determine the Y-coordinate of Vertex A**:
- Since vertex A is directly above point C, we add the length of AC to the y-coordinate of point C:
- The y-coordinate of C is -3.
- Therefore, the y-coordinate of A = y-coordinate of C + length of AC = -3 + 12 = 9.

### Final Answer:
The y-coordinate of vertex A is **9**.