Constructing a parallel through a point
(translated triangle method)
Geometry construction using a compass and straightedge
This page shows how to construct a line
to a given line through a given point with compass and straightedge or ruler.
This construction works by creating any triangle between the given point and the
given line, then copying
that triangle any distance along the given line. Since we know that a translation
can map the one triangle onto the second
then the lines linking the corresponding
points of each triangle are parallel, and we can create the desired parallel line by linking the top
vertices of the two triangles.
Printable step-by-step instructions
The above animation is available as a
printable step-by-step instruction sheet, which can be used for making handouts
or when a computer is not available.
This construction works by creating a triangle and then translating (sliding) the triangle along the given line.
All corresponding vertices of a translated polygon are linked by lines that are congruent and parallel.
This can be seen more clearly
in the animation at
Translating a Polygon.
(In that animation, check the "Show Lines" box ).
||Triangle ARB and A'R'B' are congruent
||By construction. A'R'B' was copied from ARB. For method and proof see
Copying a triangle
||AB and A'B' are
||All four points lie on PQ
||A'R'B' is a translation of ARB.
||The two triangles are congruent
(from 1) and not
(from 2) and not
reflected (by construction).
||RR' is parallel to AA'
||Lines linking the corresponding vertices of translated polygons are parallel.
See Properties of translated polygons
||RR' is parallel to PQ
||From (2) - AA' is parallel to PQ because they are
Try it yourself
Click here for a printable parallel line construction worksheet containing two 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.
Thanks to Sue Connolly, Irondequoit High School, New York for contributing this construction.
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Other constructions pages on this site
Circles, Arcs and Ellipses
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