Geometry Geometry Constructions & Loci 2 — Questions and Answers
Question 1: The locus of all points at a fixed distance from a given line is:
- Two parallel lines, one on each side of the given line (Correct answer)
- A circle centered on the line
- A single line parallel to the given line
- A perpendicular bisector
Correct answer: Two parallel lines, one on each side of the given line
All points equidistant from a line form two parallel lines, one on each side at that fixed distance.
Question 2: Which construction produces a line through a given point parallel to a given line?
- Copying the angle formed by a transversal to create equal corresponding angles (Correct answer)
- Drawing the perpendicular bisector of the given line
- Bisecting the angle between the point and the line
- Constructing a circle at the given point
Correct answer: Copying the angle formed by a transversal to create equal corresponding angles
By copying the angle a transversal makes with the given line at the new point, you create equal corresponding angles, producing a parallel line.
Question 3: What is the locus of all points equidistant from two parallel lines?
- A circle centered between them
- A third line parallel to and midway between them (Correct answer)
- The perpendicular bisector of both lines
- Two intersecting lines
Correct answer: A third line parallel to and midway between them
The locus of points equidistant from two parallel lines is a single line parallel to both and midway between them.
Question 4: To construct an equilateral triangle on a given segment AB, you:
- Measure 60° at each endpoint and draw lines until they meet
- Set compass to length AB, draw arcs from A and B, connect intersection to A and B (Correct answer)
- Draw perpendicular bisector and measure up the same length
- Bisect the segment and draw a vertical line
Correct answer: Set compass to length AB, draw arcs from A and B, connect intersection to A and B
Place the compass at A with radius AB and draw an arc, repeat from B; their intersection is the third vertex, and all sides will equal AB.
Question 5: What is the locus of the vertex of a right angle whose two sides pass through two fixed points?
- A straight line
- An ellipse
- A circle with the two fixed points as endpoints of a diameter (Correct answer)
- A parabola
Correct answer: A circle with the two fixed points as endpoints of a diameter
By Thales' theorem, any angle inscribed in a semicircle is a right angle, so the locus is a circle with the two points as diameter endpoints.
Question 6: How do you construct a perpendicular to a line through a point ON the line?
- Draw two arcs from the point along the line, then bisect the resulting segment above the line (Correct answer)
- Measure 90° with a protractor at the point
- Draw a circle at the point and find where it crosses the line
- Copy the angle adjacent to the point
Correct answer: Draw two arcs from the point along the line, then bisect the resulting segment above the line
Place the compass at the point, mark two equidistant points on the line, then construct the perpendicular bisector of that segment.
Geometry Geometry Constructions & Loci 2
The locus of all points at a fixed distance from a given line is: