Perpendicular is the Shortest Theorem

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Here we will prove that of all the straight lines that can
be drawn to a straight line from a given point outside it, the perpendicular is
the shortest.

Given: XY is a straight line and O is a point outside it. OP
is perpendicular to XY and OZ is an oblique.

Perpendicular is the Shortest

To Prove: OP < OZ.


Proof:

            Statement

            Reason

1. In ∆OPZ, ∠OPZ = 90°.

1. OP ⊥ XY.

2. ∠OZP is an acute angle.

2. In a triangle, if one angle is a right angle, the other two must be acute.

3. ∠OZP < ∠OPZ.

3. From statement 1 and 2.

4. OP < OZ. (proved)

4. In a triangle, the greater angle has the greater side opposite to it.

9th Grade Math

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