A is a point on the y-axis whose ordinate is 5 and B = (-3,1) .Find AB.

Answer

To find the distance AB between the points A and B, we will follow these steps:

### Step 1: Identify the coordinates of points A and B.
- Point A is on the y-axis with an ordinate of 5, which means its coordinates are \( A(0, 5) \).
- Point B is given as \( B(-3, 1) \).

### Step 2: Use the distance formula.
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Step 3: Substitute the coordinates into the formula.
Here, \( A(0, 5) \) gives us \( x_1 = 0 \) and \( y_1 = 5 \), and \( B(-3, 1) \) gives us \( x_2 = -3 \) and \( y_2 = 1 \). Substituting these values into the distance formula:
\[ d = \sqrt{((-3) - 0)^2 + (1 - 5)^2} \]

### Step 4: Simplify the expression.
Calculating the squares:
\[ d = \sqrt{(-3)^2 + (-4)^2} \]
\[ d = \sqrt{9 + 16} \]
\[ d = \sqrt{25} \]

### Step 5: Calculate the final distance.
Taking the square root:
\[ d = 5 \]

Thus, the distance \( AB \) is \( 5 \).