Construct a Perpendicular through a Point on a Line

In this presentation, Chris Tisdell delves into the world of geometric construction, specifically focusing on the use of a circle arc template as a versatile tool that can replace traditional compasses in geometry education. The circle arc template, which is part of a larger mathematical tool, offers greater accuracy and safety when drawing circle arcs. Tisdell begins by introducing the audience to this critical tool.

He shares his screen, revealing the circle arc template, its markings, and its unique capabilities. The tool allows for precise positioning of circle arcs by aligning its centre with a designatedpoint and using a straight edge in conjunction. Tisdell emphasises that while the template contains various markings, not all of them are necessary for theconstructions he demonstrates.

The main geometric problem explored in this presentation involves constructing a perpendicular line segment to a given line segment that passes through a specified point on the given segment. This is a classic problem in geometry that has been studied for centuries. The challenge here is to achieve this construction accurately and safely, which is where the circle arc template comes into play.

Tisdell begins by drawing a line segment AB and labelling its endpoints as A and B. He aims to construct a line perpendicular to AB through a point C located on AB. To demonstrate this, he uses the circle arc template to draw an extended arc that intersects AB at two new points, D and E. The midpoint of DE, labeled as C, serves as the desired point of intersection for the perpendicular line segment. 

The construction of an equilateral triangle with DE as its base is the next step in solving this problem. Tisdell refers to a previous video where he demonstrated how to construct an equilateral triangle with a given base using the same circle arc template and straight edge.However, in this case, he wants to create the equilateral triangle with base AB instead.

Using the circle arc template, Tisdell carefully constructs two arcs from points D and E, each intersecting the other side of DE. These intersections yield two new points, F and G, completing the equilateral triangle with base DE.

To ensure the perpendicularity of the constructed line segment IC to AB, Tisdell points out that triangle CGE and triangle CIF are congruent by side-side-side congruence. This congruence demonstrates that angle DCI is equal to angle ECI, thereby proving that line segment IC is indeed perpendicular to AB at point C.

In summary, Chris Tisdell's presentation showcases the utility of the circle arc template in solving classic geometric problems with precision and safety. His step-by-step construction of a perpendicular line segment through a given point on a line segment serves as an excellent example of how this tool can enhance geometry education and geometric problem-solving.