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Perpendicular at a point on a line
Geometry construction using a compass and straightedge
Step-by-step Instructions Printer friendly version
After doing this Your work should look like this
Start with a line and point K on that line. Geometry construction with compass and straightedge or ruler or ruler
1.  Set the compass width to a medium setting. The actual width does not matter. Geometry construction with compass and straightedge or ruler or ruler
2.  Without changing the compass width, mark a short arc on the line at each side of the point K, forming the points P,Q. These two points are thus the same distance from K. Geometry construction with compass and straightedge or ruler or ruler
3.  With the compass on P, set its width to any setting beyond K. Geometry construction with compass and straightedge or ruler or ruler
4.  Mark off an arc on one side of the line. Geometry construction with compass and straightedge or ruler or ruler
5.  Without changing the compass width repeat from the point Q so that the the two arcs cross each other, creating the point R Geometry construction with compass and straightedge or ruler or ruler
6.  Using the straight edge, draw a line from K to where the arcs cross.
7.  Done. The line KR just drawn is a perpendicular to the line PQ at K  

Proof

This construction works by effectively building two congruent triangles. The image below is the final drawing above with the red lines added.

  Argument Reason
1 Segment KP is congruent to KQ They were both drawn with the same compass width
2 Segment PR is congruent to QR They were both drawn with the same compass width
3 Triangles ∆KRP and ∆KRQ are congruent Three sides congruent (SSS). KR is common to both.
4 Angles PKR, QKR are congruent CPCTC. Corresponding parts of congruent triangles are congruent
5 Angles PKR QKR are both 90° They are a linear pair and (so add to 180°) and congruent (so each must be 90°)

  - Q.E.D
Try it yourself
Click here for a printable construction worksheet containing two 'perpendiculars from a point' problems to try. When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.

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