--- title: "Constructions and Loci" description: "Use compasses and a straightedge to construct angle bisectors, perpendicular bisectors and perpendiculars. Describe loci and shade regions that meet distance conditions." canonical: https://lightmysky.com/learn/mathematics/constructions-and-loci-mt_yqoXwrKaW3 source: https://lightmysky.com/learn/mathematics/constructions-and-loci-mt_yqoXwrKaW3.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Constructions and Loci Use compasses and a straightedge to construct angle bisectors, perpendicular bisectors and perpendiculars. Describe loci and shade regions that meet distance conditions. Subject: Mathematics · Area: Geometry · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/constructions-and-loci-mt_yqoXwrKaW3 ## Ready when they can - Construct angle bisectors, perpendicular bisectors and perpendiculars, leaving the construction arcs visible - Describe loci set by distances from points, lines and line segments - Shade the region that satisfies two conditions at once ## Lesson: Bisectors, loci and shading regions An angle bisector splits an angle into two equal angles. Draw an arc centred at the vertex, crossing the arms at D and E. From D and E, draw equal-radius arcs meeting inside the angle. Join their meeting point to the vertex with a straightedge. The later setting may differ from the first arc's setting. Divide the whole angle by 2 to find a half; double a half to recover the whole. Bisecting a half again divides by 2 again. *(drawing: Follow these construction stages. Leave the arcs visible so their settings and intersections can be checked.)* A perpendicular bisector crosses a segment at its midpoint and at 90 degrees. Set the compass wider than half the segment. From both endpoints, use that same setting to draw arcs meeting on both sides of the segment. Join the two meetings. Every point on this line is equally far from the endpoints. Conversely, every point equally far from the endpoints lies on this line. Matching arc settings make the intersections equidistant. **Example.** Find a coordinate midpoint by averaging each coordinate separately. For A(1, 3) and B(7, 5), x = (1 + 7) / 2 = 4 and y = (3 + 5) / 2 = 4. Their midpoint is (4, 4). The perpendicular bisector passes through it. Other points can lie on the perpendicular bisector too. To construct a perpendicular through P on or off a line, draw an arc centred at P crossing the line at two distinct points A and B. For P off the line, use a large enough radius. From A and B, draw arcs of equal radius greater than half AB, meeting on both sides. Join the intersections. This perpendicular passes through P because PA = PB. Keep all arcs to show the method. A ray inside a right angle splits its 90 degrees into two angles. A locus is the set of all points satisfying a rule. Points exactly 2 cm from P form a circle boundary centred at P, with radius 2 cm. Points within 2 cm form the disc: the boundary and its interior. Points closer than 2 cm form only the interior. Here, within includes equality; closer than excludes it. Distance is the shortest distance. Within d cm of an infinite line gives a strip reaching d cm on each side, of width 2d. For a segment, use the perpendicular distance if its foot lies on the segment; otherwise use the nearest endpoint. Exactly d cm from a segment, for d > 0, forms a stadium boundary: two parallel segments joined by end semicircles. Within d cm includes its interior too. For rules joined by and, keep only their overlap. Closer to A than B means A's side of the perpendicular bisector of AB, excluding the bisector. Within 4 cm of C gives C's disc, including its boundary. Q's side of m is an open half plane: exclude m, and Q must not lie on m. If m passes through a circle's centre, points strictly inside the circle on Q's side form a semicircular interior. Exclude its diameter and curved boundary. **Recap.** Use matching compass settings for paired arcs. Check each distance rule and its boundary, then keep only points satisfying every condition. *Lesson, questions and flashcards written for this dataset.* ## Practice 17 questions on this page, each with its working shown.