|
|
Perpendicular at a point on a line
Geometry construction using a compass and straightedge
Click on 'Next' to go through the construction one step at a time, or click on 'Run' to let it run without stopping.
(If there is no image below, see support page.)
| After doing this |
Your work should look like this |
| Start with a line and point K on that line. |
 |
| 1. Set the compass width to a medium setting. The actual width does not matter. |
 |
| 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. |
 |
| 3. Increase the compass to almost double its width (again the exact setting is not important). |
 |
| 4. From P, mark off a short arc above K |
 |
| 5. Without changing the compass width repeat from the point Q so that the the two arcs cross each other, creating the point R |
 |
| 6. Using the straight edge, draw a line from K to where the arcs cross. |
 |
| 7. Done. The line just drawn is a perpendicular to the line 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.
Constructions pages on this site
Lines
Angles
Triangles
Triangle Centers
Circles, Arcs and Ellipses
Non-Euclidean constructions
(C) 2009 Copyright John Page
|
|
