I’m revising for a test and my compass and I are on a slightly wobbly adventure. I need to construct the locus of points that are the same distance from two lines that cross each other. I know this should be a neat, precise construction with straightedge and compass, but my brain keeps cartwheeling.
Here’s my totally wrong attempt: I measured the gap between the lines at one place with a ruler, halved it, and then drew a line halfway between them as if the lines were parallel. That gave me a very smug-looking strip… but then I realized the distance between non-parallel lines isn’t constant, so yeah, that’s nonsense. I also tried drawing a circle centered at the intersection point and hoped it would somehow be the right locus. It wasn’t. Oops.
Could someone explain the correct construction steps to get the locus of points equidistant from two intersecting lines, using only compass and straightedge? And follow-up question: if the two lines are parallel instead, what does the locus look like and how would I construct it then? Bonus tiny confusion: if I’m only given segments of the lines (not the full infinite lines), do the segment endpoints affect the locus I should draw?
Thank you! I promise to stop measuring the wrong thing soon.
Ready to make maths more enjoyable, accessible, and fun? Join a friendly community where you can explore puzzles, ask questions, track your progress, and learn at your own pace.
By becoming a member, you unlock:
3 Responses
For two lines that intersect, the locus of points equidistant from them is the pair of angle bisectors-both the one inside the acute angle and the one inside the obtuse angle. Construction-wise: put the compass on the intersection point and draw an arc that cuts both lines; from those two cut-points, draw equal-radius arcs that cross each other; then draw the line from the vertex through that crossing point. That’s one bisector. Repeat on the other “side” to get the second bisector. A small side note from my own compass chronicles: I once drew a big circle centered at the intersection and declared victory… which was adorable but wrong. The circle isn’t the whole locus, but the points where it meets the bisectors are on the locus-so it’s a nice way to spot-check that your bisector really is splitting the angle. Also, I used to think only the acute-angle bisector “counts” because it sits between the two arms, but actually both bisectors are valid for the equidistance condition.
If the lines are parallel, the locus is a single line: the midline exactly halfway between them, parallel to both. One way to construct it is to draw a common perpendicular that cuts both lines (you can do this by constructing a perpendicular to one line through any point and extending until it hits the other), take the midpoint of that segment, and then through that midpoint draw a line parallel to the originals. Do that at two different places if you like, then join the midpoints-same result. When I was learning this, I kept drawing a whole “strip” as if the locus were every point in between; it looks satisfying, but only the middle line has equal distances to both parallels. The boundary lines themselves aren’t part of the locus unless the distance between the parallels is zero, which would be… well, they’d be the same line, oops.
About being given only segments: in most school problems they still mean the infinite lines, and you just draw the angle bisectors (or the single midline) and show the relevant portion across the picture. The endpoints don’t change the locus. If the question truly means “segments” as objects you measure distance to, things get quirky near the ends-once you’re past the foot of the perpendicular, the nearest point on a segment is an endpoint, and then parts of the locus can bend into little circular bits around those endpoints. But for standard “equidistant from two lines” tasks, ignore the segment endpoints and stick with the two angle bisectors (intersecting case) or the single halfway-parallel (parallel case).
For two intersecting lines, the locus is the pair of angle bisectors: from the intersection draw an arc cutting both lines, then with the same radius draw arcs from those cut points to meet, and join the vertex to that meeting point (repeat for the other angle)-it’s like creasing a page corner so the edges line up.
If the lines are parallel, the locus is the single midline parallel to both-draw any common perpendicular, midpoint the segment between the lines, and through it draw a line parallel to them (think walking exactly down the middle of a hallway); with only segments, this is unchanged if you mean the containing lines, but if you truly mean the finite segments then only the portions between perpendiculars through the segment endpoints count.
Yes! The locus for intersecting lines is the pair of angle bisectors-swing equal arcs from the intersection to mark points on each line and join matching marks; for example, if the lines meet at 60°, the bisectors lie at 30° and 120°. For parallel lines, drop a perpendicular across them, take its midpoint, and through it draw a line parallel to both (the midline); if you only have finite segments, just stop the bisectors at perpendiculars through the endpoints and bridge the gaps with small circular arcs centered at the endpoints.