Ideas in Geometry/Constructions I

Trickier Constructions - Marcus Eck

Introduction

This is clearly one of the "trickier" sections that is covered in this textbook. To master this section, simple memorization is not required. Rather, one must apply techniques learned in previous chapters and use problem-solving strategies and thought.

The main idea of this section is constructing shapes when you are only given a limited amount of information. In section 4.1 (Constructions) we learned some basic constructions that are very necessary in our section. To stand a chance in section 4.2, one must learn the following constructions from 4.1 first: Equilateral triangle, copying a segment, bisecting a segment, constructing a perpendicular line, bisecting an angle, copying an angle, constructing a parallel line and constructing lines that are tangent to a circle (this one is not as crucial as the others).

Once you have memorized these constructions and have a good understanding of why they work you are ready for some tricky constructions.

Let's walk through one of the first examples of a tricky construction.

We are asked to construct a triangle when we are only given the three sides. We do not know any of the angles, only the sides.

So what steps should we take to begin this process? The book clearly lays out how to complete this construction step by step, but lets try to do this by ourselves using our minds.

A smart step for any construction is to copy down what we are given and know. We are given sides A,B and C but we do not know how they are connected, so a good place to start is to copy the first segment, AB. We know this is in the right place, now how do we get the other two in the correct location? We could copy the other two segments repeatedly and see where they match up...but that isn't too exact and would take a very long time. This is thinking along the right track though, how can we do this in a more precise measure? What shape can we use that has the same length to all of its points? A circle! If we construct a circle at point A with radius of length AC this shows us all the possibilities where segment AC and point C can end up. Now lets do the same thing at point B with segment BC. Now we have two circles on our paper, one with a center at point A with a radius of AC and one with its center at point B with a radius of BC (which shows us all the possibilities, again, of where point C can end up). So we have two circles that are showing where point C could be. How do we know where it actually is? If you said where they intersect, you are correct! Point C cannot be in two places at once, so the intersection point(s) of these two circles is where point C must lie. Since the circles intersect in two places, you must choose which point you want to be C. Notice, though, that each point creates the same triangle, just reversed on itself. To complete the problem, all you have to do is connect the dots.

Here is an image that may help you through the process: Complete Triangle

Final Words

There is no clearly laid out solution for these problems and they are complex. We need to discover some ways that will improve our efficiency and effectiveness with problems such as these. I have to emphasize page 127 in the book. The other pages only provide practice and examples, but the bulk of the chapter lies on this ever-important page. These are the strategies you can use to give you the ability to solve these problems on your own.

"Construct what you can. You should start by constructing anything you can, even if you don't se how it will help you with your final construction. In doing so you are "chipping away" at the problem just as a rock-cutter ships away at a large boulder." - We did this in out example problem when we copied the first segment, AB. It opened the problem up and enabled us to think more clearly about our next step.

More guidelines as stated in the book

1) If a side is given, then you should draw it

2) If an angle is given and you know where to put it, draw it

3) If an altitude of length L is given, then draw a line parallel to the side that the altitude is perpendicular to. The new line must be distance L from the side. Lets break this one down a little farther as to make it less complex. If an altitude is given, this can also mean a height, draw a parallel line to the side the altitude is perpendicular to. This means that if their is a height given, you should draw a line that is parallel to the opposite side of the height. This will make the picture more clear and will create many distances that are all of length L.

4) If a median is given, then bisect the segment it connects to and draw a circle centered around the bisector, whose radius is the length of the median.

5) If you are working on a figure, construct any "mini-figures" inside the figure you are trying to construct. For example, many of the problems below ask you to construct a triangle. Some of these constructions have right-triangles inside of them, which are easier to construct than the final figure.

Using your mind and these techniques, these constructions are solvable despite how impossible they may seem.