Geometric constructions with compass and straightedge

About this lesson

Integrated Math I › Transformations and congruence

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Geometric constructions with compass and straightedge

Warm-up

Three quick questions from earlier lessons.

Discover

With only a compass and a straightedge (no ruler marks), can you find the exact midpoint of a segment?

  1. Yes, with two equal arcs
  2. No, you need to measure

Key idea

Constructions use equal distances, not measurements.

  1. Step 1The angle at O.
  2. Step 21. An arc centered at O crosses the sides at A and B.
  3. Step 32. Equal arcs from A and from B meet at P.
  4. Step 43. The ray OP bisects the angle (SSS: OAP ≅ OBP).

A compass keeps a distance fixed; a straightedge draws lines through points you already have. With them you can copy a segment or an angle, bisect a segment or an angle, draw perpendicular and parallel lines, and inscribe a regular hexagon (step the radius 6 times around the circle), an equilateral triangle (every other hexagon vertex) or a square (two perpendicular diameters). Each step is justified by congruent triangles or equal radii.

Worked example

Bisect the angle at vertex O.

it crosses the sides at A and B, with OA = OB
Arc from O.
they meet at P, with AP = BP
Equal arcs from A and B.
triangles OAP and OBP are congruent (SSS), so OP bisects the angle
Draw OP.

Your turn, guided

Your turn: a regular hexagon inside a circle of radius 5 cm. Each side?

55 cm
The side equals the radius.

Practice ★

  1. Put the steps for constructing the perpendicular bisector of a segment AB of length 7 cm in order.
    • With the compass point on A, draw an arc with a radius of more than half of AB
    • Keep the same opening; with the point on B, draw an arc that crosses the first one twice
    • Label the two crossing points P and Q
    • Draw the line through P and Q with the straightedge
  2. Put the steps for constructing the perpendicular bisector of a segment AB of length 9 cm in order.
    • With the compass point on A, draw an arc with a radius of more than half of AB
    • Keep the same opening; with the point on B, draw an arc that crosses the first one twice
    • Label the two crossing points P and Q
    • Draw the line through P and Q with the straightedge
  1. To inscribe a regular hexagon in a circle of radius 8 cm, you step the compass around the circle. Which opening do you keep?
    1. 8 cm, the radius of the circle
    2. 16 cm, the diameter
    3. Any opening, as long as it never changes

Watch out!

Practice ★★

  1. In the construction of the perpendicular bisector of AB (AB = 10 cm), why does the line PQ through the two arc crossings bisect AB at a right angle?
    1. P and Q are each as far from A as from B, and such points lie on the perpendicular bisector
    2. Because the angle measures 90° with a protractor
    3. Because the arcs have a radius of 10 cm
  2. In the construction of the perpendicular bisector of AB (AB = 7 cm), why does the line PQ through the two arc crossings bisect AB at a right angle?
    1. P and Q are each as far from A as from B, and such points lie on the perpendicular bisector
    2. Because the angle measures 90° with a protractor
    3. Because the arcs have a radius of 7 cm
  1. A regular hexagon is constructed inside a circle of radius 4 cm, with its vertices on the circle. What is its perimeter, in cm?Not to scale

Exit ticket

Three questions to check you have got it.

  1. Put the steps for constructing the perpendicular bisector of a segment AB of length 7 cm in order.
    • With the compass point on A, draw an arc with a radius of more than half of AB
    • Keep the same opening; with the point on B, draw an arc that crosses the first one twice
    • Label the two crossing points P and Q
    • Draw the line through P and Q with the straightedge
  2. Put the steps for constructing the perpendicular bisector of a segment AB of length 9 cm in order.
    • With the compass point on A, draw an arc with a radius of more than half of AB
    • Keep the same opening; with the point on B, draw an arc that crosses the first one twice
    • Label the two crossing points P and Q
    • Draw the line through P and Q with the straightedge
  3. Put the steps for constructing the perpendicular bisector of a segment AB of length 6 cm in order.
    • With the compass point on A, draw an arc with a radius of more than half of AB
    • Keep the same opening; with the point on B, draw an arc that crosses the first one twice
    • Label the two crossing points P and Q
    • Draw the line through P and Q with the straightedge

Summary card

Constructions

Compass for equal distances, straightedge for lines; justify with congruent triangles.

Watch out: do not change the compass opening between the two arcs.

End of the lesson

Next lesson: Translations and rigid transformations Back to the chapter

Answers: Geometric constructions with compass and straightedge

A

  1. 1. With the compass point on A, draw an arc with a radius of more than half of AB 2. Keep the same opening; with the point on B, draw an arc that crosses the first one twice 3. Label the two crossing points P and Q 4. Draw the line through P and Q with the straightedge
  2. 1. With the compass point on A, draw an arc with a radius of more than half of AB 2. Keep the same opening; with the point on B, draw an arc that crosses the first one twice 3. Label the two crossing points P and Q 4. Draw the line through P and Q with the straightedge

B

  1. 8 cm, the radius of the circle

C

  1. P and Q are each as far from A as from B, and such points lie on the perpendicular bisector
  2. P and Q are each as far from A as from B, and such points lie on the perpendicular bisector

D

  1. 2424

E

  1. 1. With the compass point on A, draw an arc with a radius of more than half of AB 2. Keep the same opening; with the point on B, draw an arc that crosses the first one twice 3. Label the two crossing points P and Q 4. Draw the line through P and Q with the straightedge
  2. 1. With the compass point on A, draw an arc with a radius of more than half of AB 2. Keep the same opening; with the point on B, draw an arc that crosses the first one twice 3. Label the two crossing points P and Q 4. Draw the line through P and Q with the straightedge
  3. 1. With the compass point on A, draw an arc with a radius of more than half of AB 2. Keep the same opening; with the point on B, draw an arc that crosses the first one twice 3. Label the two crossing points P and Q 4. Draw the line through P and Q with the straightedge

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