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Using Properties of Perpendicular Lines

Concept

Perpendicular Lines

A pair of lines that intersect at a right angle are said to be perpendicular.

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Exercise

Name all pairs of perpendicular lines in the diagram.

Show Solution
Solution

We can begin by explicitly noting the given information in the diagram. First, we know that — they aren't perpendicular. Also given is that which is our first pair of perpendicular lines. This information about and can help us to decide if and are perpendicular. As the angles at the intersection of and are corresponding to the ones at the intersection of and

Thus, and are also a pair of perpendicular lines. Lastly, is perpendicular to any of the other lines? Looking at the diagram, we can see that it's not. However, this can also be argued mathematically. If was true, would also be parallel to and Similarly, if it was perpendicular to and it would be parallel to As intersects all other lines, it's not parallel to any of them. Thus, it can't possibly be perpendicular to any line either.

Example

Conclusion

There are two pairs of perpendicular lines in the diagram: and

Rule

Perpendicular Postulate

The perpendicular postulate states that for a point not on the line there exists only one line perpendicular to that passes through

This postulate is one of the very basic truths used when proving and finding further characteristics of perpendicular lines.

Construction

Drawing a Perpendicular Line

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Given a point on the line it is possible to use a compass and straightedge to create a line that is perpendicular to and that passes through This will be the unique line described by the Perpendicular Postulate.

First, place the sharp end of the compass at point and draw a circle.
Draw circle

The circle intersects at two distinct points. These points will be named and for later referencing.

Place the sharp end of the compass at and draw an arc above point
Draw arc

Next, using the same compass setting, the sharp end is placed at and another arc is drawn above such that it intersects the first arc.
Draw arc

By using a straightedge to draw a line through and the intersection of the arcs, the desired perpendicular line is constructed.
Draw Line

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Rule

Shortest Distance from a Point to a Line

The shortest distance from a point to a line is always along the line perpendicular to through

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Exercise

Find the shortest distance from the point to the line

Show Solution
Solution

The shortest route from to is along the line through perpendicular to To find the desired distance, we have to first find where this perpendicular line intersects Perpendicular lines, expressed as linear functions, satisfy the following condition. Thus, the slope of shortest route from to is the negative reciprocal of the slope of Let's find the slope of Increasing by increases the value of by Therefore, the slope of is giving us the desired slope We can now draw a line through with the slope

These two lines intersect at so the shortest distance from to is the distance from at to We'll calculate this using the distance formula.

The shortest distance between and is units.

Concept

Perpendicular Bisector

The perpendicular bisector of a segment is the line that is perpendicular to and intersects at its midpoint.

Construction

Drawing a Perpendicular Bisector

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Given the segment a compass and straightedge can be used to draw its perpendicular bisector.

Place the compass' sharp end at one of the segment's endpoints. Draw an arc with a radius larger than half the distance between and

Keeping the compass length the same, draw a corresponding arc on the opposite side. The two arcs should now intersect at two distinct points.

The line that contains these two intersections can now be drawn using a straightedge.

This line is perpendicular to and their intersection is at the midpoint of Thus, it is the desired perpendicular bisector.

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