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Wize High School Geometry Textbook (Common Core) > Foundations of Geometry

Length of a Line Segment

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Length of a Line Segment


Given a line segment with endpoints (x1, y1)(x_1,~y_1) and (x2, y2)(x_2,~y_2), the exact length of the line segment is the distance between the two endpoints:
d=(x2−x1)2+(y2−y1)2\boxed{\Large{d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}}}


Wize Tip
Always label your points carefully! Most mistakes happen because of unlabelled or mislabelled points!

You get to pick which endpoint is (x1, y1)(x_1,\ y_1) and which is (x2, y2)(x_2,\ y_2).

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Example: Length of a Line Segment


Find the distance between (4,5)\left(4,5\right) and (−4, 9)\left(-4,\ 9\right)

First label the points

x1y1(4,5)\begin{array}{ccccc} &\colorTwo{x_1}&&\colorTwo{y_1}\\ (&\colorTwo{4}&,&\colorTwo{5}&) \end{array} and x2y2(−4,9)\begin{array}{ccccc} &\colorThree{x_2}&&\colorThree{y_2}\\ (&\colorThree{-4}&,&\colorThree{9}&) \end{array}

Using the distance formula:

d=(x2−x1)2+(y2−y1)2d=\sqrt{(\colorThree{x_2}-\colorTwo{x_1})^2+(\colorThree{y_2}-\colorTwo{y_1})^2}

d=(−4−4)2+(9−5)2d=\sqrt{(\colorThree{-4}-\colorTwo{4})^2+(\colorThree{9}-\colorTwo{5})^2}

d=(−8)2+(4)2d=\sqrt{(-8)^2+(4)^2}

d=64+16d=\sqrt{64+16}

d=80d=\sqrt{80}

d≈8.944d\approx8.944

ANSWER: d=80\boxed{d=\sqrt{80}} or d≈ 8.944\boxed{d\approx\ 8.944}

Practice: Length of a Line Segment

Find the distance between (4,5)\left(4,5\right) and (−6, 7)\left(-6,\ 7\right).

Practice: Length of a Line Segment

A dog at the point (1,−2)(1,-2) has a leash that is tethered to a post located at (−4,2)(-4,2). Find the area that this dog can cover while still tethered to this post.

Practice: Length of a Line Segment

Find the shortest distance between the point (4,3)(4,3) and the line y=−12x+52y=-\dfrac{1}{2}x+\dfrac{5}{2}.