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

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

$(1, - 3)$

Step 1

Convert

$2 \frac{1}{2}$ to an improper fraction.

A mixed number is an addition of its whole and fractional parts.

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

Add

$2$ and

$\frac{1}{2}$ .

To write

$2$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

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

Combine

$2$ and

$\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply

$2$ by

$2$ .

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

Add

$4$ and

$1$ .

$(- \frac{5}{2}, - 3) - (1, - 3)$

Step 2

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 3

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

$\sqrt{{(1 - (- \frac{5}{2}))}^{2} + {((- 3) - (- 3))}^{2}}$

Step 4

Simplify.

Multiply

$- (- \frac{5}{2})$ .

Multiply

$- 1$ by

$- 1$ .

$\sqrt{{(1 + 1 (\frac{5}{2}))}^{2} + {((- 3) - (- 3))}^{2}}$

Multiply

$\frac{5}{2}$ by

$1$ .

$\sqrt{{(1 + \frac{5}{2})}^{2} + {((- 3) - (- 3))}^{2}}$

Write

$1$ as a fraction with a common denominator.

$\sqrt{{(\frac{2}{2} + \frac{5}{2})}^{2} + {((- 3) - (- 3))}^{2}}$

Combine the numerators over the common denominator.

$\sqrt{{(\frac{2 + 5}{2})}^{2} + {((- 3) - (- 3))}^{2}}$

Add

$2$ and

$5$ .

$\sqrt{{(\frac{7}{2})}^{2} + {((- 3) - (- 3))}^{2}}$

Apply the product rule to

$\frac{7}{2}$ .

$\sqrt{\frac{7^{2}}{2^{2}} + {((- 3) - (- 3))}^{2}}$

Raise

$7$ to the power of

$2$ .

$\sqrt{\frac{49}{2^{2}} + {((- 3) - (- 3))}^{2}}$

Raise

$2$ to the power of

$2$ .

$\sqrt{\frac{49}{4} + {((- 3) - (- 3))}^{2}}$

Multiply

$- 1$ by

$- 3$ .

$\sqrt{\frac{49}{4} + {(- 3 + 3)}^{2}}$

Add

$- 3$ and

$3$ .

$\sqrt{\frac{49}{4} + 0^{2}}$

Raising

$0$ to any positive power yields

$0$ .

$\sqrt{\frac{49}{4} + 0}$

Add

$\frac{49}{4}$ and

$0$ .

$\sqrt{\frac{49}{4}}$

Rewrite

$\sqrt{\frac{49}{4}}$ as

$\frac{\sqrt{49}}{\sqrt{4}}$ .

$\frac{\sqrt{49}}{\sqrt{4}}$

Simplify the numerator.

Rewrite

$49$ as

$7^{2}$ .

$\frac{\sqrt{7^{2}}}{\sqrt{4}}$

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

$\frac{7}{\sqrt{4}}$

Simplify the denominator.

Rewrite

$4$ as

$2^{2}$ .

$\frac{7}{\sqrt{2^{2}}}$

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

$\frac{7}{2}$

Step 5

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

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