Distance between two point `(8, -4)` and `(0, a)` is `10`. All the values are in the same unit of length. Find the positive value of a.

Explanation

To find the positive value of \( a \) given the points \( (8, -4) \) and \( (0, a) \) with a distance of \( 10 \), we can follow these steps:

### Step 1: Use the Distance Formula
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
In our case, the points are \( (8, -4) \) and \( (0, a) \), and the distance is \( 10 \).

### Step 2: Substitute the Points into the Formula
Substituting the coordinates into the distance formula, we have:
\[ 10 = \sqrt{(0 - 8)^2 + (a - (-4))^2} \]
This simplifies to:
\[ 10 = \sqrt{(-8)^2 + (a + 4)^2} \]
\[ 10 = \sqrt{64 + (a + 4)^2} \]

### Step 3: Square Both Sides
To eliminate the square root, we square both sides:
\[ 10^2 = 64 + (a + 4)^2 \]
\[ 100 = 64 + (a + 4)^2 \]

### Step 4: Isolate the Squared Term
Now, we isolate the squared term:
\[ 100 - 64 = (a + 4)^2 \]
\[ 36 = (a + 4)^2 \]

### Step 5: Take the Square Root
Taking the square root of both sides gives us:
\[ \sqrt{36} = a + 4 \]
This results in two equations:
\[ a + 4 = 6 \quad \text{and} \quad a + 4 = -6 \]

### Step 6: Solve for \( a \)
1. From \( a + 4 = 6 \):
\[ a = 6 - 4 = 2 \]
2. From \( a + 4 = -6 \):
\[ a = -6 - 4 = -10 \]

### Step 7: Identify the Positive Value
The positive value of \( a \) is:
\[ \boxed{2} \]