Find the Distance Between Two Points (1,4) , (9,16)

$(1, 4)$ , $(9, 16)$

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.

$\sqrt{{(9 - 1)}^{2} + {(16 - 4)}^{2}}$

Step 3

Simplify.

Subtract $1$ from $9$ .

$\sqrt{8^{2} + {(16 - 4)}^{2}}$

Raise $8$ to the power of $2$ .

$\sqrt{64 + {(16 - 4)}^{2}}$

Subtract $4$ from $16$ .

$\sqrt{64 + 12^{2}}$

Raise $12$ to the power of $2$ .

$\sqrt{64 + 144}$

Add $64$ and $144$ .

$\sqrt{208}$

Rewrite $208$ as $4^{2} \cdot 13$ .

Factor $16$ out of $208$ .

$\sqrt{16 (13)}$

Rewrite $16$ as $4^{2}$ .

$\sqrt{4^{2} \cdot 13}$

Pull terms out from under the radical.

$4 \sqrt{13}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$4 \sqrt{13}$

Decimal Form:

$14.42220510 \dots$

Step 5