# Ruler and Compass

## GeoGebra tasks

Allowed tools |
---|

New Point |

Intersect Two Objects |

Line through Two Points |

Segment between Two Points |

Ray through Two Points |

Circle with Centre through Point |

Compasses |

**Construction 1** Bisect angle

** Start with:** Two rays with a common endpoint.

** Construct:** A ray dividing the angle between the rays
in two equally large angles, such a ray is called an angle **bisector**;
dividing into two equal parts is called **bisecting**.

**Construction 2** Bisect segment

** Start with:** A segment.

** Construct:** The midpoint.

**Construction 3** Perpendicular line 1

** Start with:** A line and a point on the line.

** Construct:** A line through the point perpendicular
to the given line, a so called **normal** line.

**Construction 4** Perpendicular line 2

** Start with:** A line and a point not lying on the line.

** Construct:** A line through the point perpendicular
to the given line.

**Construction 5** Reflect point in line

** Start with:** A line and a point.

** Construct:** The mirror image of the point when reflected
in the line. Show the traces of the original point and the mirror image
of the point by right-clicking on each point and choose ```
Trace
on
```

. (Use the construction of Perpendicular line 2.)

**Construction 6** Parallel line

** Start with:** A line and a point not on the line.

** Construct:** A line through the point parallel to the
given line. (Use the construction of Perpendicular line 2.)

**Construction 7** Multiple of a segment

** Start with:** A segment.

** Construct:** A segment who's length is a multiple of
the length of the given segment. (For example three times as long.)

In addition to the previous tools you can now also use:

More allowed tools |
---|

Midpoint or Centre |

Perpendicular Bisector |

Perpendicular Line |

Angle Bisector |

Parallel Line |

Reflect Object in Line |

**Construction 8** Copy circle

** Start with:** A circle `c` and a point `P`.

** Construct:** A new circle having the same radius as
the circle `c` and having the centre `P`.

You can use the construction above to copy a segment of a given length. You can now use these tools as well:

Even more allowed tools |
---|

Circle with Centre and Radius |

Segment with Given Length from Point |

**Construction 9** Copy angle

** Start with:** Two rays `a` and `b`
with a common endpoint `P`, and a ray `c` having another
endpoint `Q`.

** Construct:** A ray `d` with the endpoint `Q`
such that the angle between `c` and `d` is equal to
the angle between `a` and `b`.

**Construction 10** Dividing
a segment into three equal parts (trisecting)

** Start with:** A segment`. `

** Construct:** Trisect the segment.

**Hint:** Start the construction as below.

**Comment**: The task of
trisecting an **angle** would have been just a tad more difficult.

## Proofs

Use the three congruence theorems, the theorem about isosceles triangles and the theorems about angles to prove that the constructions 1-7 are correct.

Hint: Place triangles in appropriate ways into the constructions.

### Example, Construction 1 Bisect angle

You must show that the angles `BAC` and `CAD`
are equal.

Mark the triangles `ΔABC` and `ΔACD`.

Use one of the congruence theorems to show that the triangles are congruent. Justify that you can use this theorem.

Show that the angles `BAC` and `CAD` are equal.

### Example, Construction 2 Bisect segment

You must show that the segments `AE` and `EB`
are equal.

Show this for the angles: `DAB=DBA=CAB=CBA`.

Then show this for the angles: `BDC=ADC=ACD=BCD` .

Show that `ΔAED≅ΔBED`!

by Malin Christersson under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Sweden License