Perpendicular is the Shortest
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.
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. |
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