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

Answer

To solve the problem, we need to find the value of \( x \) such that the point \( A(x, 2) \) is equidistant from the points \( B(8, -2) \) and \( C(2, -2) \). Then we will calculate the length of \( AB \).

### Step 1: Set up the distance equations
Since point \( A \) is equidistant from points \( B \) and \( C \), we can set the distances \( AB \) and \( AC \) equal to each other.

The distance \( AB \) can be calculated using the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
For points \( A(x, 2) \) and \( B(8, -2) \):
\[ AB = \sqrt{(8 - x)^2 + (-2 - 2)^2} = \sqrt{(8 - x)^2 + (-4)^2} = \sqrt{(8 - x)^2 + 16} \]

Now, calculate the distance \( AC \):
For points \( A(x, 2) \) and \( C(2, -2) \):
\[ AC = \sqrt{(2 - x)^2 + (-2 - 2)^2} = \sqrt{(2 - x)^2 + (-4)^2} = \sqrt{(2 - x)^2 + 16} \]

### Step 2: Set the distances equal
Now we set \( AB \) equal to \( AC \):
\[ \sqrt{(8 - x)^2 + 16} = \sqrt{(2 - x)^2 + 16} \]

### Step 3: Square both sides
To eliminate the square roots, we square both sides:
\[ (8 - x)^2 + 16 = (2 - x)^2 + 16 \]

### Step 4: Simplify the equation
Subtract 16 from both sides:
\[ (8 - x)^2 = (2 - x)^2 \]

Now, expand both sides:
\[ (8 - x)(8 - x) = (2 - x)(2 - x) \]
\[ 64 - 16x + x^2 = 4 - 4x + x^2 \]

### Step 5: Cancel \( x^2 \) and simplify further
Subtract \( x^2 \) from both sides:
\[ 64 - 16x = 4 - 4x \]

Now, rearranging gives:
\[ 64 - 4 = 16x - 4x \]
\[ 60 = 12x \]

### Step 6: Solve for \( x \)
Now divide both sides by 12:
\[ x = \frac{60}{12} = 5 \]

### Step 7: Find the length of \( AB \)
Now that we have \( x = 5 \), we can find the length of \( AB \):
\[ AB = \sqrt{(8 - 5)^2 + (-2 - 2)^2} = \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Final Answer
The value of \( x \) is \( 5 \) and the length of \( AB \) is \( 5 \).

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