Maths

332 MATHEMATICS
Remark : The example of the proof above shows you, yet again, that there can be
several ways of proving a result.
Theorem A1.2 : Out of all the line segments, drawn from a point to points of a
line not passing through the point, the smallest is the perpendicular to the line.
Proof :
Fig. A1.5
Statements
Analysis/Comment
Let XY be the given line, P a point not lying on XY Since we have to prove that
and PM, PA 1
, PA 2
, . . . etc., be the line segments out of all PM, PA 1
, PA 2
, . . .
drawn from P to the points of the line XY, out of etc., the smallest is perpendiwhich
PM is the smallest (see Fig. A1.5). cular to XY, we start by
taking these line segments.
Let PM be not perpendicular to XY
Draw a perpendicular PN on the line XY, shown
by dotted lines in Fig. A1.5.
PN is the smallest of all the line segments PM,
PA 1
, PA 2
, . . . etc., which means PN < PM.
This contradicts our hypothesis that PM is the
smallest of all such line segments.
Therefore, the line segment PM is perpendicular
to XY.
This is the negation of the
statement to be proved by
contradiction.
We often need
constructions to prove our
results.
Side of right triangle is less
than the hypotenuse and
known property of numbers.
Precisely what we want!
We reach the conclusion.