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?
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
 Here is square \(ABCD\) with diagonal \(BD\) drawn:
 Construct a circle centered at \(A\) with radius \(AD\).
 Construct a circle centered at \(C\) with radius \(CD\).
 Draw the diagonal \(AC\) and write a conjecture about the relationship between the diagonals \(BD\) and \(AC\).
 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?
 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:
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.