Find the distance between the points (-5,3) and (3,1) :

Options

A. `2sqrt7`
B. `3sqrt(14)`
C. `5sqrt(17)`
D. `2sqrt(17)`

Answer

To find the distance between the points (-5, 3) and (3, 1), we will use the distance formula derived from the Pythagorean theorem. The distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

### Step 1: Identify the coordinates
Let the first point be \((x_1, y_1) = (-5, 3)\) and the second point be \((x_2, y_2) = (3, 1)\).

### Step 2: Substitute the coordinates into the formula
Now we will substitute the values into the distance formula:

\[ d = \sqrt{(3 - (-5))^2 + (1 - 3)^2} \]

### Step 3: Simplify the expressions inside the square root
Calculating the differences:

1. For \(x\)-coordinates:
\[ 3 - (-5) = 3 + 5 = 8 \]

2. For \(y\)-coordinates:
\[ 1 - 3 = -2 \]

Now substituting these values back into the formula:

\[ d = \sqrt{(8)^2 + (-2)^2} \]

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

1. \(8^2 = 64\)
2. \((-2)^2 = 4\)

Now we can substitute these values:

\[ d = \sqrt{64 + 4} \]

### Step 5: Add the results
Adding the results inside the square root:

\[ d = \sqrt{68} \]

### Step 6: Simplify the square root
To simplify \(\sqrt{68}\), we can factor it:

\[ \sqrt{68} = \sqrt{4 \times 17} = \sqrt{4} \times \sqrt{17} = 2\sqrt{17} \]

### Final Result
Thus, the distance between the points (-5, 3) and (3, 1) is:

\[ d = 2\sqrt{17} \]

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