20-August-2008
Perpendiculars, Construction and Properties
A. Ruler and Compass Construction of a Perpendicular
from a Point P to a line L
Step 1: Draw a circle centered at P which meets the line segment at two points, say A and B.
Triangle APB is isosceles:
Step 2: Draw a second triangle AQB isometric to triangle APB, base AB, and vertex Q on the "other" side of line segment AB using the ruler and compass based SSS triangle construction criteria:
Next draw the line segment PQ and its point of intersection D with the line segment AB.
That completes the construction.
Claim: Line segment PD is perpendicular to the original line L
Proof: Observe triangles PQA and PQB are isometric by the SSS criteria. Further they are isoceles. Hence the four angles at P and Q in these two triangles are equal.
Now the ASA criteria implies the four triangles ADP, BDP, ADQ and PDQ are isometric. Hence the four angles at D sum to four right angles (360 degrees) and are equal. So each angle at D is a right angle. Moreover, sides AD and DB are equal in length. Hence line segment PD is a perpendicular from P to the line that includes A and B.
B. Properties of a Perpendicular from a point to a line:
Uniqueness Property: Let line segment PD be perpendicular to a line with
D a point on the line. Let C be another point on the line. Then the
distance of P to C is the length of the hypotenuse with PD as a side.
Furthermore, the angle C must be less than a right angle. So a perpendicular PD
from P to the line cannot pass through C. Hence a perpendicular to the
line from a point is unique, and thus may called the perpendicular. The end
point is unique as well.
Distance Properties and Distance of a point to a line:
Since the length of a hypotenuse is greater than the lengths of the other two sides, the distance of P to C is greater than the distance of P to D. Thus the intersection of a line with a perpendicular from a point P gives the point D closest to any point P on the line. That length or distance is called the height h of P above the line if the line is drawn horizontal, More generally, it may be called the distance of P to the line.
Properties of Circles Centered at P.
If a circle centered at P
- has radius less than the length of PD than the circle does not intersect the line.
- passes through a point C on the line different from D then the circle radius is more than the length of the line segment PD.
- has radius equal to the length of PD then the circle intersects the line at only one point, namely D, and its radius at D, the segment PD, is perpendicular to the line. There is one motivation for saying the line is tangent to the circle at D.
Let h the height of a point P above a line, or the distance of P to the line. In general, a circle centered at P will have no point of intersection with the line if its radius r < h, exactly one point of intersection if its radius r = h and two points of intersection if its radius r > h. Here those two points of intersection will be within distance r of the point D. The ratio to radius r of the distances q of the points of intersection to D approaches 1 as r approaches infinity (gets larger and larger. See next figure.
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