- Let’s use tools to create shapes precisely.
1.1: The Right Tool
Copy this figure using only the Pen tool and no other tools.
Familiarize yourself with your digital straightedge and compass tools by drawing a few circles of different sizes, drawing a few line segments of different lengths, and extending some of those line segments in both directions.
Copy the figure by completing these steps with the Line, Segment, and Ray tools and the Circle and Compass tools:
- Draw a point and label it \(A\).
- Draw a circle centered at point \(A\) with a radius equal to length \(PQ\).
- Mark a point on the circle and label it \(B\).
- Draw another circle centered at point \(B\) that goes through point \(A\).
- Draw a line segment between points \(A\) and \(B\).
1.2: Illegal Construction Moves
- Create a circle centered at \(A\) with radius \(AB\).
- Estimate the midpoint of segment \(AB\), mark it with the Point on Object tool, and label it \(C\).
- Create a circle centered at \(B\) with radius \(BC\). Mark the 2 intersection points with the Intersection tool. Label the one toward the top of the page as \(D\) and the one toward the bottom as \(E\).
- Use the Polygon tool to connect points \(A\), \(D\), and \(E\) to make triangle \(ADE\).
1.3: Can You Make a Perfect Copy?
Here is a hexagon with all congruent angles and all congruent sides (called a regular hexagon).
Try to draw a copy of the regular hexagon using only the pen tool. Draw your copy next to the hexagon given, and then drag the given one onto yours. What happened?
Here is a figure that shows the first few steps to constructing the regular hexagon. Use straightedge and compass moves to finish constructing the regular hexagon. Drag the given one onto yours and confirm that it fits perfectly onto itself.
- How do you know each of the sides of the shape are the same length? Show or explain your reasoning.
Why does the construction end up where it started? That is, how do we know the central angles go exactly 360 degrees around?
To construct geometric figures, we use a straightedge and a compass. These tools allow us to create precise drawings that someone else could copy exactly.
- We use the straightedge to draw a line segment, which is a set of points on a line with 2 endpoints.
- We name a segment by its endpoints. Here is segment \(AB\), with endpoints \(A\) and \(B\).
- We use the compass to draw a circle, which is the set of all points the same distance from the center.
We describe a circle by naming its center and radius. Here is the circle centered at \(F\) with radius \(FG\).
Early mathematicians noticed that certain properties of shapes were true regardless of how large or small they were. Constructions were used as a way to investigate what has to be true in geometry without referring to numbers or direct measurements.
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 \(O\).
- line segment
A set of points on a line with two endpoints.