Find the Distance Between Two Points (1,1/2) , (1/3,2)

$(1, \frac{1}{2})$ , $(\frac{1}{3}, 2)$

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.

$\sqrt{{(\frac{1}{3} - 1)}^{2} + {(2 - \frac{1}{2})}^{2}}$

Step 3

Simplify.

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\sqrt{{(\frac{1}{3} - 1 \cdot \frac{3}{3})}^{2} + {(2 - \frac{1}{2})}^{2}}$

Combine $- 1$ and $\frac{3}{3}$ .

$\sqrt{{(\frac{1}{3} + \frac{- 1 \cdot 3}{3})}^{2} + {(2 - \frac{1}{2})}^{2}}$

Combine the numerators over the common denominator.

$\sqrt{{(\frac{1 - 1 \cdot 3}{3})}^{2} + {(2 - \frac{1}{2})}^{2}}$

Simplify the numerator.

Multiply $- 1$ by $3$ .

$\sqrt{{(\frac{1 - 3}{3})}^{2} + {(2 - \frac{1}{2})}^{2}}$

Subtract $3$ from $1$ .

$\sqrt{{(\frac{- 2}{3})}^{2} + {(2 - \frac{1}{2})}^{2}}$

Move the negative in front of the fraction.

$\sqrt{{(- \frac{2}{3})}^{2} + {(2 - \frac{1}{2})}^{2}}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{2}{3}$ .

$\sqrt{{(- 1)}^{2} {(\frac{2}{3})}^{2} + {(2 - \frac{1}{2})}^{2}}$

Apply the product rule to $\frac{2}{3}$ .

$\sqrt{{(- 1)}^{2} \frac{2^{2}}{3^{2}} + {(2 - \frac{1}{2})}^{2}}$

Raise $- 1$ to the power of $2$ .

$\sqrt{1 \frac{2^{2}}{3^{2}} + {(2 - \frac{1}{2})}^{2}}$

Multiply $\frac{2^{2}}{3^{2}}$ by $1$ .

$\sqrt{\frac{2^{2}}{3^{2}} + {(2 - \frac{1}{2})}^{2}}$

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

$\sqrt{\frac{4}{3^{2}} + {(2 - \frac{1}{2})}^{2}}$

Raise $3$ to the power of $2$ .

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

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Combine $2$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $2$ .

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

Subtract $1$ from $4$ .

$\sqrt{\frac{4}{9} + {(\frac{3}{2})}^{2}}$

Apply the product rule to $\frac{3}{2}$ .

$\sqrt{\frac{4}{9} + \frac{3^{2}}{2^{2}}}$

Raise $3$ to the power of $2$ .

$\sqrt{\frac{4}{9} + \frac{9}{2^{2}}}$

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

$\sqrt{\frac{4}{9} + \frac{9}{4}}$

To write $\frac{4}{9}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sqrt{\frac{4}{9} \cdot \frac{4}{4} + \frac{9}{4}}$

To write $\frac{9}{4}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\sqrt{\frac{4}{9} \cdot \frac{4}{4} + \frac{9}{4} \cdot \frac{9}{9}}$

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{4}{9}$ by $\frac{4}{4}$ .

$\sqrt{\frac{4 \cdot 4}{9 \cdot 4} + \frac{9}{4} \cdot \frac{9}{9}}$

Multiply $9$ by $4$ .

$\sqrt{\frac{4 \cdot 4}{36} + \frac{9}{4} \cdot \frac{9}{9}}$

Multiply $\frac{9}{4}$ by $\frac{9}{9}$ .

$\sqrt{\frac{4 \cdot 4}{36} + \frac{9 \cdot 9}{4 \cdot 9}}$

Multiply $4$ by $9$ .

$\sqrt{\frac{4 \cdot 4}{36} + \frac{9 \cdot 9}{36}}$

Combine the numerators over the common denominator.

$\sqrt{\frac{4 \cdot 4 + 9 \cdot 9}{36}}$

Simplify the numerator.

Multiply $4$ by $4$ .

$\sqrt{\frac{16 + 9 \cdot 9}{36}}$

Multiply $9$ by $9$ .

$\sqrt{\frac{16 + 81}{36}}$

Add $16$ and $81$ .

$\sqrt{\frac{97}{36}}$

Rewrite $\sqrt{\frac{97}{36}}$ as $\frac{\sqrt{97}}{\sqrt{36}}$ .

$\frac{\sqrt{97}}{\sqrt{36}}$

Simplify the denominator.

Rewrite $36$ as $6^{2}$ .

$\frac{\sqrt{97}}{\sqrt{6^{2}}}$

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\sqrt{97}}{6}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{97}}{6}$

Decimal Form:

$1.64147630 \dots$

Step 5

Find the Distance Between Two Points (1,1/2) , (1/3,2)

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