# 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

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:

### 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.