How to Use the distance formula to find the length of a line segment

We can measure the length of a line segment or horizontal line on the coordinate plane simply by counting the coordinates; however, finding the length of a diagonal is more difficult. You can use the distance formula to find the length of a line. The distance formula is a generalized case of the Pythagorean Theorem, you can think of a given line segment as the hypotenuse of a right triangle. ^{[1] X Research Source} By applying basic formulas in geometry, it becomes quite easy to calculate the length of a line in the coordinate plane.

Table of Contents

Formula setting

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Set the distance formula. The formula states that

d

=

(

x

2

-

x

first

)

2

+

(

y

2

-

y

first

)

2

{displaystyle d={sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}}

d={sqrt {(x_{{2}}-x_{{1}})^{{2}}+(y_{{2}}-y_{{1}})^{{2}}}}

, in there,

d

{displaystyle d}

d

is the length of the line segment,

(

x

first

,

y

first

)

{displaystyle (x_{1},y_{1})}

(x_{{1}},y_{{1}})

are the coordinates of the first point of the line segment and

(

x

2

,

y

2

)

{displaystyle (x_{2},y_{2})}

(x_{{2}},y_{{2}})

are the coordinates of the second point of the line segment. ^{[2] X Research Source}

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Find the coordinates of the starting and ending points of the line segment. Usually these coordinates are given. If not, count along the horizontal and vertical axes to find the coordinates.

The horizontal axis is the horizontal axis; The vertical axis is the vertical axis.
The coordinates of a point are shown as follows $($ $x$ $,$ $y$ $)$ ${displaystyle (x,y)}$ $(x,y)$ .
For example, a line segment can be specified by the point $($ $2$ $,$ $first$ $)$ ${displaystyle(2,1)}$ $(2,1)$ and $($ $6$ $,$ $4$ $)$ ${displaystyle(6,4)}$ $(6,4)$ .

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Substitute coordinates into the distance formula. You must be careful to substitute the values for the correct variables. Two coordinates

x

{displaystyle x}

x

must be the first number in parentheses, leaving two coordinates

y

{displaystyle y}

y

is the second number in parentheses.

For example, we have points $($ $2$ $,$ $first$ $)$ ${displaystyle(2,1)}$ $(2,1)$ and $($ $6$ $,$ $4$ $)$ ${displaystyle(6,4)}$ $(6,4)$ , the formula becomes: $d$ $=$ $($ $6$ $-$ $2$ $)$ $2$ $+$ $($ $4$ $-$ $first$ $)$ $2$ ${displaystyle d={sqrt {(6-2)^{2}+(4-1)^{2}}}}$ $d={sqrt {(6-2)^{{2}}+(4-1)^{{2}}}}$

Calculate the length

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Calculate subtraction in parentheses. According to the rules for the order in which operations are performed, any operations enclosed in parentheses must be solved first.

For example:
$d$ $=$ $($ $6$ $-$ $2$ $)$ $2$ $+$ $($ $4$ $-$ $first$ $)$ $2$ ${displaystyle d={sqrt {(6-2)^{2}+(4-1)^{2}}}}$ $d={sqrt {(6-2)^{{2}}+(4-1)^{{2}}}}$
$d$ $=$ $($ $4$ $)$ $2$ $+$ $($ $3$ $)$ $2$ ${displaystyle d={sqrt {(4)^{2}+(3)^{2}}}}$ $d={sqrt {(4)^{{2}}+(3)^{{2}}}}$

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Square the value in parentheses. According to the rules for the order of operations, the next step is to calculate the exponent.

For example:
$d$ $=$ $($ $4$ $)$ $2$ $+$ $($ $3$ $)$ $2$ ${displaystyle d={sqrt {(4)^{2}+(3)^{2}}}}$ $d={sqrt {(4)^{{2}}+(3)^{{2}}}}$
$d$ $=$ $16$ $+$ $9$ ${displaystyle d={sqrt {16+9}}}$ $d={sqrt {16+9}}$

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Add the terms below the radical sign. Perform the math as you would for integers.

For example:
$d$ $=$ $16$ $+$ $9$ ${displaystyle d={sqrt {16+9}}}$ $d={sqrt {16+9}}$
$d$ $=$ $25$ ${displaystyle d={sqrt {25}}}$ $d={sqrt {25}}$

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Find the answer $d$ ${displaystyle d}$ $d$ . To get the final result, calculate the root of the sum of the terms in the radical sign.

Since we’re looking for the square root, you may need to round your answer.
Since this is a problem on the coordinate plane, the answer will be “common units”, not centimeters, meters or other metric units.
For example:
$d$ $=$ $25$ ${displaystyle d={sqrt {25}}}$ $d={sqrt {25}}$
$d$ $=$ $5$ ${displaystyle d=5}$ $d=5$

Advice

Avoid confusing this formula with other formulas such as the formula for finding the coordinates of the midpoint, calculating the slope of a line, the equation of a straight line, or a linear equation.
Remember the order in which calculations are performed when doing math. Subtract first, then square the differences, then add, and finally find the square root.

Steps

Formula setting

Calculate the length

Advice

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