If the point A(a,2) is equidistant from the points B(8,-2) and C(2,-2) find the value of a.

Answer

To solve the problem, we need to find the value of \( a \) such that the point \( A(a, 2) \) is equidistant from points \( B(8, -2) \) and \( C(2, -2) \). We will use the distance formula to set up our equation.

### Step-by-Step Solution:

1. **Write the Distance Formula**:
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

2. **Calculate Distance AB**:
The distance \( AB \) between points \( A(a, 2) \) and \( B(8, -2) \) is:
\[ AB = \sqrt{(8 - a)^2 + (-2 - 2)^2} \]
Simplifying this:
\[ AB = \sqrt{(8 - a)^2 + (-4)^2} = \sqrt{(8 - a)^2 + 16} \]

3. **Calculate Distance AC**:
The distance \( AC \) between points \( A(a, 2) \) and \( C(2, -2) \) is:
\[ AC = \sqrt{(2 - a)^2 + (-2 - 2)^2} \]
Simplifying this:
\[ AC = \sqrt{(2 - a)^2 + (-4)^2} = \sqrt{(2 - a)^2 + 16} \]

4. **Set Distances Equal**:
Since \( A \) is equidistant from \( B \) and \( C \), we have:
\[ AB = AC \]
Therefore:
\[ \sqrt{(8 - a)^2 + 16} = \sqrt{(2 - a)^2 + 16} \]

5. **Square Both Sides**:
To eliminate the square roots, we square both sides:
\[ (8 - a)^2 + 16 = (2 - a)^2 + 16 \]

6. **Simplify the Equation**:
We can cancel \( 16 \) from both sides:
\[ (8 - a)^2 = (2 - a)^2 \]

7. **Expand Both Sides**:
Expanding both sides gives:
\[ (8 - a)(8 - a) = (2 - a)(2 - a) \]
This simplifies to:
\[ 64 - 16a + a^2 = 4 - 4a + a^2 \]

8. **Cancel \( a^2 \)**:
The \( a^2 \) terms on both sides cancel out:
\[ 64 - 16a = 4 - 4a \]

9. **Rearrange the Equation**:
Move all terms involving \( a \) to one side and constant terms to the other:
\[ 64 - 4 = 16a - 4a \]
This simplifies to:
\[ 60 = 12a \]

10. **Solve for \( a \)**:
Divide both sides by \( 12 \):
\[ a = \frac{60}{12} = 5 \]

### Final Answer:
The value of \( a \) is \( 5 \).