# Tutorial:Practice Block I

## Square Construction

In this activity you are going to use the following tools. Make sure you know how to use each tool before you begin with the actual construction of the square:

Segment | |

Perpendicular Line | |

Line | |

Circle With Center Through Point | |

Intersect | |

Polygon | |

Show / Hide Object | |

Move |

### Construction Steps

- Draw segment a = AB between points A and B
- Construct perpendicular line b to segment AB through point B
- Construct circle c with center B through point A
- Intersect circle c with perpendicular line b to get intersection point C
- Construct perpendicular line d to segment AB through point A
- Construct circle e with center A through point B
- Intersect perpendicular line d with circle e to get intersection point D
- Create polygon ABCD (Don’t forget to close the polygon by clicking on point A after selecting point D.)
- Hide circles and perpendicular lines
- Perform the drag test to check if your construction is correct

## Regular Hexagon Construction

In this activity you are going to use the following tools. Make sure you know how to use each tool before you begin with the actual construction of the hexagon:

Circle With Center Through Point | |

Intersect | |

Polygon | |

Angle | |

Show / Hide Object | |

Move |

### Construction Steps

- Draw a circle with center A through point B
- Construct another circle with center B through point A
- Intersect the two circles in order to get the vertices C and D.
- Construct a new circle with center C through point A.
- Intersect the new circle with the first one in order to get vertex E.
- Construct a new circle with center D through point A.
- Intersect the new circle with the first one in order to get vertex F.
- Construct a new circle with center E through point A.
- Intersect the new circle with the first one in order to get vertex G.
- Draw hexagon FGECBD.
- Create the angles of the hexagon.
- Perform the drag test to check if your construction is correct.

## Circumscribed Circle of a Triangle

In this activity you are going to use the following tools. Make sure you know how to use each tool before you begin with the actual construction of the circumscribed circle:

Polygon | |

Perpendicular Bisector | |

Intersect | |

Circle With Center Through Point | |

Move |

### Construction Steps

- Create an arbitrary triangle ABC
- Construct the line bisector for each side of the triangle. The tool
*Line bisector*can be applied to an existing segment. - Create intersection point D of two of the line bisectors. The tool
*Intersect two objects*can’t be applied to the intersection of three lines. Either select two of the three line bisectors successively, or click on the intersection point and select one line at a time from the appearing list of objects in this position. - Construct a circle with center D through one of the vertices of triangle ABC
- Perform the drag test to check if your construction is correct.

Modify your construction to answer the following questions:

- Can the circumcenter of a triangle lie outside the triangle? If yes, for which types of triangles is this true?
- Try to find an explanation for using line bisectors in order to create the circumcenter of a triangle.

## Visualize the Theorem of Thales

In this activity you are going to use the following tools. Make sure you know how to use each tool before you begin with the actual construction:

Segment | |

Semicircle through 2 Points | |

Point | |

Polygon | |

Angle | |

Move |

### Construction Steps

- Draw segment AB
- Construct a semicircle through points A and B. The order of clicking points A and B determines the direction of the semicircle.
- Create a new point C on the semicircle. Check if point C really lies on the arc by dragging it with the mouse.
- Create triangle ABC
- Create the interior angles of triangle ABC

Try to come up with a graphical proof for this theorem.

**Hint:**Create midpoint O of segment AB and display the radius OC as a segment.