Construction Techniques 4: Parallel and Perpendicular Lines
- Let’s use tools to draw parallel and perpendicular lines precisely.
6.1: Math Talk: Transformations
Each pair of shapes is congruent. Mentally identify a transformation or sequence of transformations that could take one shape to the other.
6.2: Standing on the Shoulders of Giants
Here is a line \(m\) and a point \(C\) not on the line. Use straightedge and compass tools to construct a line perpendicular to line \(m\) that goes through point \(C\). Be prepared to share your reasoning.
The line segment \(AB\) has a length of 1 unit. Construct its perpendicular bisector and draw the point where this line intersects our original segment \(AB\). How far is this new point from \(A\)?
We now have 3 points drawn. Use a pair of points to construct a new perpendicular bisector that has not been drawn yet and label its intersection with segment \(AB\). How far is this new point from \(A\)?
If you repeat this process of drawing new perpendicular bisectors and considering how far your point is from A, what can you say about all the distances?
6.3: Parallel Constructions Challenge
Here is a line \(m\) and a point \(C\) not on the line. Use straightedge and compass moves to construct a line parallel to line \(m\) that goes through point \(C\).
When we write the instructions for a construction, we can use a previous construction as one of the steps. We now know 2 new constructions that are made up of a sequence of moves.
- Perpendicular lines are lines that meet at a 90 degree angle.
- Parallel lines are lines that don’t intersect. One way to make parallel lines is to draw 2 lines perpendicular to the same line.
- 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.