# 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.