4.6: Distance from a Point to a Line
- Page ID
- 34140
Suppose we are given a point \(P\) and a line \(\overleftrightarrow{AB}\) as in Figure \(\PageIndex{1}\). We would like to find the shortest line segment that can be drawn from \(P\) to \(\overleftrightarrow{AB}\).
First we will prove a theorem:
In a right triangle, the hypotenuse is larger than either leg. In Figure \(\PageIndex{1}\), \(c>a\) and \(c>b\). (The symbol ">" means "is greater than.")
- Proof
-
By the Pythagorean Theorem,
\(c = \sqrt{a^2 + b^2} > \sqrt{a^2} = a.\)
\(c = \sqrt{a^2 + b^2} > \sqrt{b^2} = b.\)
Now we can give the answer to our question:
The perpendicular is the shortest line segment that can be drawn from a point to a straight line.
In Figure \(\PageIndex{3}\) the shortest line segment from \(P\) to \(\overleftrightarrow{AB}\) is \(PD\). Any other line segment, such as \(PC\), must be longer.
- Proof
-
\(PC\) is the hypotenuse of right triangle \(PCD\). Therefore by Theorem \(\PageIndex{1}\), \(PC > PD\).
We define the distance from a point to a line to be the length of the perpendicular.
Find the distance from \(P\) to \(\overleftrightarrow{AB}\):
Solution
Draw \(PD\) perpendicular to \(\overleftrightarrow{AB}\) (Figure \(\PageIndex{4}\)). \(\triangle PCD\) is a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle.
\(\begin{array} {rcl} {\text{hyp}} & = & {2s} \\ {8} & = & {2(CD)} \\ {4} & = & {CD} \\ {L} & = & {s\sqrt{3}} \\ {PD} & = & {4\sqrt{3}} \end{array}\)
Answer: \(4\sqrt{3}\)
Problems
1 - 6. Find the distance from \(P\) to \(\overleftrightarrow{AB}\):
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