 Perpendiculars from points ON and OFF a given line MathBitsNotebook.com Terms of Use   Contact Person: Donna Roberts  Use only your compass and straight edge when drawing a construction. No free-hand drawing!

 Perpendicular from a point ON a line.
 Given: point P on a given line Construct: a line through P perpendicular to given line. STEPS:
1. Place your compass point on P and swing an arc of any size below the line that crosses the line twice. You will be drawing at least a semicircle. (Note: While you can draw this arc above or below the line, below the arc keeps the construction lines from bumping into one another.)
2. Stretch the compass LARGER!
3. Place the compass point where the arc crossed the line on one side and make a small arc above the line (the arc could be below the line if you prefer).
4.
Without changing the span on the compass, place the compass point where the first arc crossed the line on the OTHER side and make another arc. Your two small arcs should be intersecting.
5. Using a straightedge, connect the intersection of the two small arcs to point P. Does this construction look familiar?  Proof of Construction:
This construction is actually another version of the construction of BISECT AN ANGLE. This construction bisected the straight angle P. Since a straight angle contains 180º, this construction created two 90º angles. Since two right angles have been formed, a perpendicular was created.

 Perpendicular from a point OFF a line.
 Given: point P off a given line Construct: a line through P perpendicular to given line. STEPS:
1. Place your compass point on P and swing an arc of any size that crosses the line twice.
2. Place the compass point on one of the two locations where the arc crossed the line and make a small arc below the line (on the side where P is not located).
3. Without changing the span on the compass, place the compass point on the other location where the first arc crossed the line and make another small arc below the line. The two small arcs should be intersecting (on the side of the line opposite of point P).
4. Using a straightedge, connect the intersection of the two small arcs to point P. Does this construction look familiar? The construction method shown above on the right can also be applied to the construction where P is located ON the line. Proof of Construction: Label the construction: A and B are the points of intersection with the first arc, C is the intersection of the two smaller arcs, and D is the intersection of the perpendicular with the given line. Draw .
PA = PB
and AC = BC since they were constructed as radii of the same circles. These segments are also congruent. Using as a common side, ΔPBC is congruent to ΔPAC by SSS. By CPCTC, ∠APC is congruent to ∠BPC. Now, ΔAPD is congruent to ΔBPD by SAS, with common side . By CPCTC, ∠PDA is congruent to ∠PDB. These two angles form a linear pair making them supplementary. If two angles are both congruent and supplementary, they are right angles. Consequently, .  