Calculate the distance between the points (6,-4) and (3,2) correct to 2 decimal places.

Answer

To calculate the distance between the points (6, -4) and (3, 2), we will use the distance formula. The distance formula is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Where:
- \( (x_1, y_1) \) = (6, -4)
- \( (x_2, y_2) \) = (3, 2)

### Step 1: Identify the coordinates
We have:
- \( x_1 = 6 \)
- \( y_1 = -4 \)
- \( x_2 = 3 \)
- \( y_2 = 2 \)

### Step 2: Substitute the coordinates into the formula
Substituting the values into the distance formula:

\[ d = \sqrt{(3 - 6)^2 + (2 - (-4))^2} \]

### Step 3: Simplify the expressions
Calculating the differences:

\[ d = \sqrt{(-3)^2 + (2 + 4)^2} \]

This simplifies to:

\[ d = \sqrt{(-3)^2 + (6)^2} \]

### Step 4: Calculate the squares
Calculating the squares:

\[ d = \sqrt{9 + 36} \]

### Step 5: Add the results
Adding the results:

\[ d = \sqrt{45} \]

### Step 6: Simplify the square root
We can simplify \( \sqrt{45} \):

\[ \sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5} \]

### Step 7: Calculate the approximate value
Now, we calculate the approximate value of \( 3\sqrt{5} \). Knowing that \( \sqrt{5} \approx 2.236 \):

\[ 3\sqrt{5} \approx 3 \times 2.236 = 6.708 \]

### Step 8: Round to two decimal places
Finally, rounding \( 6.708 \) to two decimal places gives us:

\[ d \approx 6.71 \]

Thus, the distance between the points (6, -4) and (3, 2) is approximately **6.71 units**.