**** Suppose point *A* is located at (1,3)1,3 on a coordinate plane. If *AB* is 10 and the *x*-coordinate of point *B* is 9, explain how to use the Distance Formula to find the *y*-coordinate of point *B*.

To find the \( y \)-coordinate of point \( B \), we can use the Distance Formula. The Distance Formula is used to calculate the distance between two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in a coordinate plane, and is given by:

\[
AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

### Given Information:

- Point \( A \) is located at \( (1, 3) \).
- Distance \( AB = 10 \).
- The \( x \)-coordinate of point \( B \) is \( 9 \).

We need to find the \( y \)-coordinate of point \( B \), denoted as \( y_2 \).

### Step-by-Step Solution:

1. **Assign Coordinates to Points:**
   - Coordinates of \( A \): \( (x_1, y_1) = (1, 3) \)
   - Coordinates of \( B \): \( (x_2, y_2) = (9, y_2) \)

2. **Set Up the Distance Formula:**

   Substitute the known values into the Distance Formula:

   \[
   10 = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
   \]

   \[
   10 = \sqrt{(9 - 1)^2 + (y_2 - 3)^2}
   \]

3. **Simplify the Equation:**

   Calculate \( (9 - 1)^2 \):

   \[
   10 = \sqrt{8^2 + (y_2 - 3)^2}
   \]

   \[
   10 = \sqrt{64 + (y_2 - 3)^2}
   \]

4. **Square Both Sides to Eliminate the Square Root:**

   \[
   10^2 = 64 + (y_2 - 3)^2
   \]

   \[
   100 = 64 + (y_2 - 3)^2
   \]

5. **Isolate \( (y_2 - 3)^2 \):**

   \[
   100 - 64 = (y_2 - 3)^2
   \]

   \[
   36 = (y_2 - 3)^2
   \]

6. **Solve for \( y_2 \):**

   Take the square root of both sides:

   \[
   \sqrt{36} = \sqrt{(y_2 - 3)^2}
   \]

   \[
   6 = |y_2 - 3|
   \]

   This gives two possible solutions:

   \[
   y_2 - 3 = 6 \quad \text{or} \quad y_2 - 3 = -6
   \]

7. **Find the Possible Values of \( y_2 \):**

   - If \( y_2 - 3 = 6 \):
     
     \[
     y_2 = 6 + 3
     \]
     
     \[
     y_2 = 9
     \]

   - If \( y_2 - 3 = -6 \):
     
     \[
     y_2 = -6 + 3
     \]
     
     \[
     y_2 = -3
     \]

### Conclusion:

The \( y \)-coordinate of point \( B \) can be either \( \boxed{9} \) or \( \boxed{-3} \).

Would you like further details or have any questions?

### Related Questions:

1. How would you find the coordinates of a point equidistant from two given points?
2. What are the possible coordinates for point \( B \) if \( AB \) is a different distance, say \( 5 \) units instead of \( 10 \)?
3. How do you find the midpoint of a line segment connecting two points?
4. Can the Distance Formula be used in three-dimensional space, and if so, how?
5. How would the problem change if both the \( x \)-coordinate and the \( y \)-coordinate of point \( B \) were unknown?

**Tip:** Always remember to check both positive and negative roots when using the square root property, as they can provide multiple solutions in geometry problems.

Learn how to find the y-coordinate of point B using the Distance Formula. Detailed step-by-step solution with given coordinates of point A and the x-coordinate of point B. Suitable for students in Grades 7-9.

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