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Perpendicular to a line from an external point
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.)
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After doing this |
Your work should look like this |
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Start with a line and point R which is not on that line. |
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| Step 1 |
Place the compass on the given external point R. |
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| Step 2 |
Set the compass width to a approximately 50% more than the distance to the line.
The exact width does not matter. |
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| Step 3 |
Draw an arc across the line on each side of R, making sure not to adjust the compass width in between.
Label these points P and Q |
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| Step 4 |
At this point, you can adjust the compass width. Recommended: leave it as is.
From each point P,Q, draw an arc below the line so that the arcs cross. |
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| Step 5 |
Place a straightedge between R and the point where the arcs intersect.
Draw the perpendicular line from R to the line, or beyond if you wish. |
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| Step 6 |
Done. This line is perpendicular to the first line and passes through the point R. It
also bisects the segment PQ (divides it into two equal parts) |
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Proof
This construction works by adding points that define a kite. Recall that the diagonals of a kite intersect at right angles.
See Kite definition.
The image below is the final drawing above with the red lines added.
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Argument |
Reason |
| 1 |
Segment RP is congruent to RQ |
They were both drawn with the same compass width |
| 2 |
Segment PS is congruent to QS |
They were both drawn with the same compass width |
| 3 |
RPSQ is a kite |
A kite is a
quadrilateral with two distinct pairs of equal adjacent sides. |
| 4 |
RS is perpendicular to PQ |
The diagonals of a kite always intersect at right angles.
See Kite definition. |
- Q.E.D
Try it yourself
Click here for a printable construction worksheet containing two 'perpendiculars through 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
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