Point B has coordinates ​(2​,1​). The​ x-coordinate of point A is negative 7. The distance between point A and point B is 15 units. What are the possible coordinates of point​ A?
Question content area bottom
Part 1
The possible coordinates of point A are 
  
enter your response here.

To find the possible coordinates of point A, we can use the distance formula. The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

Given:
- Point B \((x_2, y_2) = (2, 1)\)
- Point A \((x_1, y_1) = (-7, y_1)\)
- Distance \(d = 15\)

Substituting these into the distance formula:

\[
15 = \sqrt{(2 - (-7))^2 + (1 - y_1)^2}
\]

Simplify the equation:

\[
15 = \sqrt{(2 + 7)^2 + (1 - y_1)^2}
\]

\[
15 = \sqrt{9^2 + (1 - y_1)^2}
\]

\[
15 = \sqrt{81 + (1 - y_1)^2}
\]

Next, square both sides to eliminate the square root:

\[
15^2 = 81 + (1 - y_1)^2
\]

\[
225 = 81 + (1 - y_1)^2
\]

Subtract 81 from both sides:

\[
144 = (1 - y_1)^2
\]

Now, solve for \(1 - y_1\):

\[
1 - y_1 = \pm \sqrt{144}
\]

\[
1 - y_1 = \pm 12
\]

This gives us two cases:

1. \(1 - y_1 = 12\)  
   \[
   y_1 = 1 - 12 = -11
   \]
   
2. \(1 - y_1 = -12\)  
   \[
   y_1 = 1 + 12 = 13
   \]

So, the possible coordinates of point A are:

\[
\boxed{(-7, -11) \text{ and } (-7, 13)}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you find the distance between two points with different coordinates?
2. How can you derive the distance formula from the Pythagorean theorem?
3. What happens to the distance formula if both points lie on a horizontal line?
4. Can you calculate the midpoint between two given points?
5. What is the equation of a circle given a center and a radius?

**Tip:** When solving for distances in geometry, squaring both sides can help eliminate square roots, simplifying your calculations.

Point B has coordinates (2, 1). The x-coordinate of point A is negative 7. The distance between point A and point B is 15 units. What are the possible coordinates of point A?

Explore how to find the possible coordinates of Point A, given the coordinates of Point B (2, 1) and the distance between them as 15 units. This problem applies the distance formula and involves key concepts in geometry and algebra. Suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation