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Geometry: High School15 chapters | 160 lessons

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Did you know that it's possible to do math without using numbers? This is exactly what Euclid did when he showed how to solve mathematical problems by drawing them out instead of with numbers.

When you draw something accurately without the use of numbers, it is called **geometric construction**. *Euclid* showed us how these geometric constructions can be used to solve mathematical problems when you don't have numbers at your disposal. With just two tools you can bisect lines and angles as well as draw circumscribed and inscribed circles.

So what are the two tools that you need? The two tools are your *compass* and your *straightedge*. If you count the *pencil* as a tool, then you have three tools. Note that I said *straightedge* and not a *ruler*. This is because a ruler has numbers. In true geometric construction, you don't have any numbers to deal with. The *compass* I mentioned isn't the one that has a needle that always point to the North Pole, but rather the type that you use to draw circles with. This type of *compass* has one end with a *sharp point* and a *pencil point* on the other. To use this tool, you put the pointy end down and you draw your arc with the other end by rotating the compass.

You will be using the *compass* a lot to find important points. To do this, you normally place your compass at two different points and draw an arc from each point. These two arcs will intersect at your *important point*. You will repeat these steps to find more important points. The straightedge is then used to connect two or more of these important points. Let's take a look.

To *bisect a line segment* you begin with your line. Visually adjust your compass so that it covers more than half of your line segment. Place your compass at one end of the line segment and draw arcs above and below the line segment. Then move your compass to the other end of the line segment, and draw arcs above and below the line segment. The arcs that you drew should intersect at the top and at the bottom. Then take your straightedge and connect these two intersections. This new line then is your *line segment bisector*.

Now, let's try *bisecting an angle*. First, place your compass at the *vertex* of your angle, the tip of your angle. Draw an arc that crosses both arms of your angle. The width of your compass is not important as long as its not too big. Now take your compass and move it to one of the intersections you just created. Draw an arc in the opening of your angle. Move your compass to the other intersection, and draw an arc in the opening of your angle. These two arcs should intersect. Take your straightedge and draw a line that connects your angle vertex with your newest intersection. This new line is your angle bisector.

You have just bisected a line and an angle without the use of numbers. You didn't even have to know how long the line segment is or how large the angle is. You didn't have to use numbers anywhere in the problem.

Let's review what you've learned now. When you draw something accurately without the use of numbers, it is called **geometric construction**. The two tools that you need to make geometric constructions are these two: *compass*, *straightedge*. You will be using your compass a lot to make various arcs from different points, these arcs will intersect at important points that you can then use the straightedge to connect with other points.

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Geometry: High School15 chapters | 160 lessons

- Line Segments & Rays: Definition & Measurement 3:59
- Types of Angles: Vertical, Corresponding, Alternate Interior & Others 10:28
- Geometric Constructions Using Lines and Angles 4:32
- Line Segment Bisection & Midpoint Theorem: Geometric Construction 4:39
- Dividing Line Segments into Equal Parts: Geometric Construction 5:22
- Parallel, Perpendicular and Transverse Lines 6:06
- Constructing Perpendicular Lines in Geometry 3:39
- Constructing an Angle Bisector in Geometry 3:36
- Methods & Tools for Making Geometric Constructions 3:17
- Constructing Equilateral Triangles, Squares, and Regular Hexagons Inscribed in Circles 5:00
- Go to High School Geometry: Introduction to Geometric Figures

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