Find the Distance Between Two Points (5,2) , (1,4)

$(5, 2)$ , $(1, 4)$

Step 1

Use the distance formula to determine the distance between the two points.

$Distance = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$

Step 2

Substitute the actual values of the points into the distance formula.

$\sqrt{{(1 - 5)}^{2} + {(4 - 2)}^{2}}$

Step 3

Simplify.

Subtract $5$ from $1$ .

$\sqrt{{(- 4)}^{2} + {(4 - 2)}^{2}}$

Raise $- 4$ to the power of $2$ .

$\sqrt{16 + {(4 - 2)}^{2}}$

Subtract $2$ from $4$ .

$\sqrt{16 + 2^{2}}$

Raise $2$ to the power of $2$ .

$\sqrt{16 + 4}$

Add $16$ and $4$ .

$\sqrt{20}$

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$\sqrt{4 (5)}$

Rewrite $4$ as $2^{2}$ .

$\sqrt{2^{2} \cdot 5}$

Pull terms out from under the radical.

$2 \sqrt{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{5}$

Decimal Form:

$4.47213595 \dots$

Step 5