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Answers without the blur. Sign up and see all textbooks for free! Q16.

Expert-verified Found in: Page 661 ### Algebra 1

Book edition Student Edition
Author(s) Carter, Cuevas, Day, Holiday, Luchin
Pages 801 pages
ISBN 9780078884801 # Find the distance between the points with the given coordinate.$\mathbf{\left(}\mathbf{2}\mathbf{,}\mathbf{3}\mathbf{\right)}\mathbf{,}\mathbf{\text{\hspace{0.17em}}}\mathbf{\left(}\mathbf{3}\mathbf{,}\mathbf{5}\mathbf{\right)}$

The distance between the coordinates $\left(2,3\right)$ and $\left(3,5\right)$ is $\sqrt{5}$ units.

See the step by step solution

## Step 1. State the concept of ‘Distance between two points in 2 dimensional plane’.

Let the two points in 2 dimensional plane be $\left({x}_{1},{y}_{1}\right)$ and $\left({x}_{2},{y}_{2}\right)$.

The distance between the points is given as:

$d=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\dots \left(1\right)$

Where $d$ is the distance.

## Step 2. Calculate the distance.

The two points which is given are $\left(2,3\right)$ and $\left(3,5\right)$.

Substitute $\left(2,3\right)$ as $\left({x}_{1},{y}_{1}\right)$ and $\left(3,5\right)$ as $\left({x}_{2},{y}_{2}\right)$ in $\left(1\right)$ to calculate the distance.

localid="1647926325268" $\begin{array}{l}d=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}}\\ =\sqrt{{\left(3-2\right)}^{2}+{\left(5-3\right)}^{2}}\\ =\sqrt{{\left(1\right)}^{2}+{\left(2\right)}^{2}}\\ =\sqrt{1+4}\\ =\sqrt{5}\end{array}$

## Step 3. State the conclusion.

Therefore, distance between the points $\left(2,3\right)$ and $\left(3,5\right)$ is width="26" style="max-width: none; vertical-align: -4px;" $\sqrt{5}$ units. ### Want to see more solutions like these? 