Lesson 7

Construction Techniques 5: Squares

  • Let’s use straightedge and compass moves to construct squares.

7.1: Which One Doesn’t Belong: Polygons

Which one doesn’t belong?

A

A green square with all sides marked 2.

B

A rectangle with sides marked 3 and 1.

C

A parallelogram with all sides marked 1.

D

An octagon with all sides marked 1.

7.2: It’s Cool to Be Square

Use straightedge and compass tools to construct a square with segment \(AB\) as one of the sides.

 

7.3: Trying to Circle a Square

  1. Here is square \(ABCD\) with diagonal \(BD\) drawn:
    1. Construct a circle centered at \(A\) with radius \(AD\).
    2. Construct a circle centered at \(C\) with radius \(CD\).
    3. Draw the diagonal \(AC\) and write a conjecture about the relationship between the diagonals \(BD\) and \(AC\).
    4. Label the intersection of the diagonals as point \(E\) and construct a circle centered at \(E\) with radius \(EB\). How are the diagonals related to this circle?

     
  2. Use your conjecture and straightedge and compass tools to construct a square inscribed in a circle.



Use straightedge and compass moves to construct a square that fits perfectly outside the circle, so that the circle is inscribed in the square. How do the areas of these 2 squares compare?

Summary

We can use what we know about perpendicular lines and congruent segments to construct many different objects. A square is made up of 4 congruent segments that create 4 right angles. A square is an example of a regular polygon since it is equilateral (all the sides are congruent) and equiangular (all the angles are congruent). Here are some regular polygons inscribed inside of circles:

 A row of growing regular polygons inscribed inside of 5 identical circles, starts with 3 sides, adds 1 side every circle and ends with 7 sides.

Glossary Entries

  • angle bisector

    A line through the vertex of an angle that divides it into two equal angles.

  • circle

    A circle of radius \(r\) with center \(O\) is the set of all points that are a distance \(r\) units from \(O\). 

    To draw a circle of radius 3 and center \(O\), use a compass to draw all the points at a distance 3 from \(O\).

  • conjecture

    A reasonable guess that you are trying to either prove or disprove.

  • inscribed

    We say a polygon is inscribed in a circle if it fits inside the circle and every vertex of the polygon is on the circle. We say a circle is inscribed in a polygon if it fits inside the polygon and every side of the polygon is tangent to the circle.

     

  • line segment

    A set of points on a line with two endpoints.

  • parallel

    Two lines that don't intersect are called parallel. We can also call segments parallel if they extend into parallel lines.

  • perpendicular bisector

    The perpendicular bisector of a segment is a line through the midpoint of the segment that is perpendicular to it. 

  • regular polygon

    A polygon where all of the sides are congruent and all the angles are congruent.