To find the possible coordinates of point A, we can use the distance formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

where $(x_1, y_1)$ are the coordinates of point A, and $(x_2, y_2)$ are the coordinates of point B. Given:

• Point B: $(x_2, y_2) = (5, 2)$
• Point A: $(x_1, y_1) = (-3, y_1)$
• Distance $d = 10$ units

Substituting these values into the distance formula:

$10 = \sqrt{(5 - (-3))^2 + (2 - y_1)^2}$

Simplify the equation:

$10 = \sqrt{(5 + 3)^2 + (2 - y_1)^2}$

$10 = \sqrt{8^2 + (2 - y_1)^2}$

$10 = \sqrt{64 + (2 - y_1)^2}$

Square both sides to eliminate the square root:

$100 = 64 + (2 - y_1)^2$

Subtract 64 from both sides:

$36 = (2 - y_1)^2$

Take the square root of both sides:

$2 - y_1 = \pm 6$

This gives us two possible cases:

1. $2 - y_1 = 6$
2. $2 - y_1 = -6$

For case 1:

$y_1 = 2 - 6 = -4$

For case 2:

$y_1 = 2 + 6 = 8$

Thus, the possible coordinates of point A are:

$\text{Point A: } (-3, -4) \text{ or } (-3, 8)$

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How would the answer change if the distance between points A and B were different?
2. What if both the x and y coordinates of point A were unknown?
3. Can you find the midpoint of the line segment joining points A and B?
4. What is the slope of the line segment connecting points A and B for both possible points A?
5. How does the distance formula relate to the Pythagorean theorem?

Tip: When solving for unknowns in geometry problems, breaking down the problem using known formulas and systematically solving step-by-step helps avoid mistakes.