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?
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:
- angle bisector
A line through the vertex of an angle that divides it into two equal angles.
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 \(O\).
A reasonable guess that you are trying to either prove or disprove.
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.
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.