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Geometry: Constructions

Perpendicular Lines

Problems

Problems

Lines (or segments) are called perpendicular if their intersection with one another forms a right angle. You can see for yourself that if one of the angles formed by the intersection of two lines or segments is a right angle, then all four angles created will also be right angles.

Figure %: Perpendicular lines
Intersecting lines that are not perpendicular to one another are called oblique lines.

Through any given line, there are an infinite number of perpendicular lines. Can you see why?

Figure %: An infinite number of lines perpendicular to any given line
Through a specific point on a line, though, there exists only one perpendicular line. Likewise, given a line and a point not on that line, there is only one perpendicular line through the noncolinear point.
Figure %: Perpendicular lines through a point on a line, and a point not on that line
In the picture on the left, line AB contains the point C. There exists only one line, line DE, that contains C and is perpendicular to line AB. In the picture on the right, point C is not on line AB. There exists only one line, line CD, that contains C and is perpendicular to line AB.

The Distance Between a Line and a Point not on that Line

When working with geometry it is a common problem to have to find the distance between a line and a point not on that line. There are many different segments that could be drawn between the point and the line, but when you need to find the distance between the point and the line, it is implied that you are seeking the shortest distance. This is found by drawing the segment through the point which is perpendicular to the line, and taking its length. The distance between a line and a noncolinear point is represented by this segment.

Figure %: The distance between a line and a point not on that line

In this figure, the shortest distance between the point C and the line AB is along the segment CD, which is perpendicular to the line AB.

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