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.