If (-3, y) is equidistant from (2, 6) and (7,-2), find y.

To find the value of \( y \) such that the point \((-3, y)\) is equidistant from the points \((2, 6)\) and \((7, -2)\), we need to 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 that the point \((-3, y)\) is equidistant from the points \((2, 6)\) and \((7, -2)\), the distances from \((-3, y)\) to each of these points must be equal.

Let's calculate the distance from \((-3, y)\) to \((2, 6)\):

\[
\sqrt{(2 - (-3))^2 + (6 - y)^2} = \sqrt{(2 + 3)^2 + (6 - y)^2} = \sqrt{5^2 + (6 - y)^2} = \sqrt{25 + (6 - y)^2}
\]

Now, let's calculate the distance from \((-3, y)\) to \((7, -2)\):

\[
\sqrt{(7 - (-3))^2 + (-2 - y)^2} = \sqrt{(7 + 3)^2 + (-2 - y)^2} = \sqrt{10^2 + (-2 - y)^2} = \sqrt{100 + (-2 - y)^2}
\]

Since the distances are equal, we can set these two expressions equal to each other:

\[
\sqrt{25 + (6 - y)^2} = \sqrt{100 + (-2 - y)^2}
\]

To eliminate the square roots, we square both sides:

\[
25 + (6 - y)^2 = 100 + (-2 - y)^2
\]

Next, expand the squared terms:

\[
25 + (36 - 12y + y^2) = 100 + (4 + 4y + y^2)
\]

Simplify both sides:

\[
25 + 36 - 12y + y^2 = 100 + 4 + 4y + y^2
\]

Combine like terms:

\[
61 - 12y + y^2 = 104 + 4y + y^2
\]

Since \( y^2 \) appears on both sides, it cancels out, leaving:

\[
61 - 12y = 104 + 4y
\]

Next, we solve for \( y \):

\[
61 - 104 = 12y + 4y
\]

\[
-43 = 16y
\]

\[
y = \frac{-43}{16} = -\frac{43}{16}
\]

Thus, the value of \( y \) is \(-\frac{43}{16}\).

Would you like any further explanation or details?

Here are some related questions you might consider:
1. How do you derive the distance formula?
2. What are some examples of using the distance formula in geometry?
3. How would you find the midpoint between two points?
4. Can you determine the equation of the line passing through \((-3, y)\) and \((2, 6)\)?
5. What is the significance of points being equidistant from two other points?

**Tip:** When solving equations involving square roots, remember to check for any extraneous solutions that might arise from squaring both sides.

Learn how to find the value of y such that the point (-3, y) is equidistant from the points (2, 6) and (7, -2) using the distance formula. This problem involves coordinate geometry concepts and is suitable for high school students in Grades 9-12.

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