Point B has coordinates of (5,2). The x-coordinate of point A is -3. The distance between point A and point B is 10 units. What are the possible coordinates of point A?

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.

Discover how to determine the possible coordinates of point A when given the distance to point B and one coordinate of point A. This problem uses the distance formula and involves key concepts in coordinate geometry. Suitable for students in Grades 9-10.

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