What you’ll learn
Construct parallel lines.
Key idea
A perpendicular to a perpendicular restores the original direction; the second perpendicular must be to the intermediate line.
The final perpendicular is to the intermediate line, and the final line must pass through the specified point.
Worked example
Enter the exact value or ordered values in the format requested below.
Line l: y=-3x+(-1); exterior point P=(0,0). Use compass/straightedge perpendicular constructions to create n through P parallel to l; m names the intermediate line. Enter exactly five semicolon-separated relations using perpendicular, belongs-to and parallel symbols; both new lines must pass P.
- P is exterior since vertical offset 1 is nonzero. Intermediate m through P has slope 1/3 perpendicular to l.
- Construct n through P perpendicular to m; slope -3, intercept 0. It is parallel and distinct from l because 0!=-1.
Answer: m⊥l; P∈m; n⊥m; P∈n; n∥l
Quick Check
Choose which candidate representation satisfies the requirements of 'Construct parallel lines' and justify the choice.
Line l: y=-3x+(3); exterior point P=(0,5). Use compass/straightedge perpendicular constructions to create n through P parallel to l; m names the intermediate line. Plan A: m through P perpendicular l, n through P perpendicular m. Plan B: both m/n through P perpendicular l. Plan C: n perpendicular m through R=(0,6) instead. Select the plan satisfying direction and incidence.