Constructions and Loci · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
به انگلیسی

Bisectors, loci and shading regions

Mathematics · Geometry · ages 15-16
Name ______________________   Date ____________

  1. A perpendicular bisector crosses segment AB at M. AM is 6.5 cm. Find MB in cm.

    Answer: ______________

  2. The first angle-bisector arc crosses the arms at D and E. What is required for the next two arcs?

    • Use equal radii from D and E.
    • Use a larger radius from D.
    • Use a larger radius from E.
    • Centre both arcs at the vertex.

  3. An 80 degree angle is bisected with compasses. What is each part?

    • 40 degrees
    • 20 degrees
    • 160 degrees

  4. Sam says the locus of points exactly 3 cm from a point P is a circle of radius 3 centred at P. Is Sam right?

    Circle one:   True   False

  5. The perpendicular from X to line AB meets segment AB at F, between A and B. XF is 2.5 cm. What is the shortest distance from X to segment AB in cm?

    Answer: ______________

  6. What is the locus of points exactly a fixed positive distance from a line segment?

    • A circular boundary
    • A straight boundary
    • A stadium boundary

  7. P is off line m. An arc centred at P marks A and B on m. Each plan below uses endpoint arcs that meet twice. Which plan constructs the perpendicular through P and leaves evidence of the method?

    • Use unequal radii from A and B; join their two intersections and keep every construction arc visible.
    • Use equal radii from A and B; join A to B instead of the intersections and keep every arc visible.
    • Use equal radii from A and B; join their two intersections and keep every arc visible.
    • Use equal radii from A and B; join their two intersections and erase every construction arc afterwards.

  8. Maya draws intersecting arcs of radius 4 cm from A and radius 5 cm from B. She joins their two intersections. Is that line the perpendicular bisector of AB?

    Circle one:   True   False

  9. A 136 degree angle is bisected, then one half is bisected again. What is the final piece in degrees?

    Answer: ______________

  10. A perpendicular n is constructed through P on line m. Ray PQ lies inside one right angle between m and n and makes 28 degrees with its boundary ray on m. Find its angle with the boundary ray on n, in degrees.

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Bisectors, loci and shading regions W1-mt_yqoXwrKaW3-s1

  1. 6.5 · M is the midpoint, so MB = AM = 6.5 cm.
  2. Use equal radii from D and E. · The arcs from D and E must share a radius and meet inside the angle. Their radius may differ from the first arc's radius.
  3. 40 degrees · Half of 80 is 40.
  4. True · Fixed distance from one point defines a circle.
  5. 2.5 · The perpendicular foot lies on the segment, so XF is its shortest distance from X: 2.5 cm.
  6. A stadium boundary · Two parallel segments join semicircles round the endpoints. Only that stadium boundary is at the exact distance.
  7. Use equal radii from A and B; join their two intersections and keep every arc visible. · Equal endpoint radii make both intersections equidistant from A and B. Joining them gives AB's perpendicular bisector. It passes through P because PA = PB. Keep the arcs as evidence of these steps.
  8. False · Each intersection is 4 cm from A and 5 cm from B. It is not equidistant from the endpoints, so the joined line is not their perpendicular bisector.
  9. 34 · 136 over 2 is 68, then 68 over 2 is 34.
  10. 62 · The two angles fill a right angle, so the required angle is 90 - 28 = 62 degrees.
Worksheet · LightMySky