Scale Factor

Here, I will be using similar polygons (circles) and showing their scale factors for units and units squared. Here are two ordinary circles, one with radius of 6, and one with radius of 10. Their scale factor is 6 divided by 10, and that simplified is 3:5.

Now when we find the circumferences of the two circles, their ratio should be the same as the ratio of the radii.

The circumferences of the two circles are 37.68 and 62.86. When we divide 37.68 by 62.86, we get .60, which is equal to 3 divided by 5. 

The areas of the circles should be in the scale factor of k squared, in other words, 3 squared / 5 squared.  The scale factor for the areas of the circles should be 9:25.

The area for the first circle is 112.99 square cm, and the area for the second circle is 314.45 square cm. If we divide 112.99 by 314.45, we get .36, which is the same as 9/25. This demonstrates that if we have any two similar polygons, any length measured in units is in the scale factor k, and anything measured in units squared is in the scale factor k squared. (~DM5)

Measuring Height Two Ways

I decided it would be neat if I were to use similar triangles to find the height of the pole. I used a tape measure to measure out 20 feet from the pole and put a marker there. Then I took a picture with my phone.  I uploaded 2 copies of the picture to GSP and made two triangles.

They are both similar but have different scales; one is real world, and the other is on GSP. GSP gave me the lengths of the triangle in centimeters, so then all I did was set up a proportion to get the length of the pole. I also had a second way to do this: I set up a tangent equation to find the pole. Both results were very close but i think that the tangent equation is more accurate. (~TR1)

Similar Triangles with Parallel Sides

 

While listening to music, I noticed that the logo on my headphones is an example of two similar triangles. To prove this, I traced both triangles, and then found the midpoints of lines AC and BC. These two points happened to land precisely where expected, and so I connected them to form a white overlay of the symbol.

I used the measuring tool in GSP to find all five angles, which are listed on the left. By using the AA postulate, we can see that triangle ABC and triangle EDC are indeed similar.

Although it's not included in the picture above, I also measured angle AED. The sum of angles AED and BAC was 180 degrees, which proves that side AB is parallel to side ED.

(~GC2)

Similar Polygons

These two books (quadrilaterals) appear similar because their sides seem to be proportional. To verify, I measured the lengths: the larger book is 10 in x 6 in, and the smaller book is 6.5 in x 4 in. But 10:6 is NOT equal to 6.5:4, so in fact they aren't similar, as the sides are not proportional. (~KR2)

Floor Tessellation: Dealing with Similar Triangles

This image shows the tile pattern on a floor, a collection of similar triangles. This assortment of triangles connects with the AA Postulate, which can be seen by looking at two small triangles. Now, since they coincide perfectly for this tessellation, you can see that the angles are congruent to each other. Since two angles are congruent, then the triangles are similar by the AA postulate.

Now take two small triangles that have connecting vertices that form vertical angles. By the SAS Similarity Theorem, these triangles have one congruent side, congruent angles (vertical), and then another pair of congruent sides. On the other hand, they also demonstrate the SSS Similarity Theorem, because all sides are proportional.

The Triangle Proportionality Theorem is also evident if you look at a larger triangle made of four smaller triangles. A line in the middle that intersects through both sides of the larger triangle is parallel to the base, which divides the sides proportionally.

The Triangle Bisector Theorem can lastly be demonstrated. If you bisect any given triangle with a ray, then it will divide the opposite side into segments proportional to the other two sides.

Overall, many theorems can be applied to this design (~JG2)