I-MATH

Similar Triangles

Please do the GSP Lab Files, Chapter 09 Similar Triangles, in connection with this web page.

Similar triangles are one of the most useful topics in geometry, in terms of applications to "real life"! Similarity is a concept that is the basis of scale drawing in architecture and engineering, used in building scale models from toy model airplanes to scale models in industry and architecture, and is very useful in measuring the heights of inaccessible objects. This type of measurement is useful for finding the heights of inaccessible buildings and mountains, and even distances in navigation.

When introducing the concept of similar triangles, it is useful to revisit "Transformations" which we discussed previously in the Triangles chapter of I-MATH. The overhead projector is perfect to demonstrate "dilation" - the original triangle on the overhead projector itself, and the projected image is the dilated triangle. If you don't have an overhead projector, then either one of the diagrams below can be drawn on the board. In each diagram, the blue triangle is the original triangle, the green triangle is the dilated image (each side is twice as long as the corresponding side of the original triangle), and point P is the center of dilation. The red "rays" show the dilation.

Architects use scale drawing to design houses, hotels, and all types of projects. The following floor plan is an example of a simple house design, originally drawn at the scale 1/4" = 1'-0". drawing is "similar" to the actual floor plan of the house: every 1/4" on the drawing would represent 1 foot in the actual house when it is built. The drawing below is reduced to fit on this web page. Scale drawings like this are essential in all aspects of architectural design and in the construction business, as well as in many other design fields, from airplane design to the design of everything from toys to cities.

The floor plan above came from a fascinating and comprehensive website on architecture, which is worth exploring beginning at the following link. This website has hundreds of links to websites on architecture:

http://architecture.about.com/arts/architecture/msubmenu4.htm

The beautiful perspective drawing below of a home in Mauna Lani, Hawaii. was designed and drawn to scale by an architectural firm called Architects Studio, whose offices are in Santa Barbara, California.

You can visit their website at

http://home1.gte.net/architek/samples.htm

These web pages will teach you how to do scale drawings:

http://www.scp-theaterworks.com/highschooltech/howto/paperwk/scale.htm

As you may recall, Connections and Applications of mathematics is one of the new NCTM Standards, as stated in the Introduction to I-MATH. the projects in this chapter relate directly to this Standard.

My students in the E-School distance learning class called Connecting Geometry use similar triangles to measure the height of a tree or building in their neighborhood, and my students at my high school use the same method to measure the height of a coconut tree on campus. Click on the link below to visit the Connecting Geometry website, and learn more about this project. Then click on the second link below, and scroll down the page to find an excellent example of student work on this topic.

http://www.k12.hi.us/~csanders/ch_09Similar.html

http://www.k12.hi.us/~csanders/SW.html

You may recall that there was a brief explanation of the fourth transformation, Dilation, in the I-MATH chapter on Symmetry and Transformations. This might be a good time to review that explanation, and redo the GSP Lab Activity on this topic, "Activity 2. 4 dilation" in the GSP Lab Files. Dilation is very directly related to similar triangles, and you might want to have your students do this GSP Lab Activity during their study of similar triangles. Again, the study of Transformational Geometry is spelled out as one of the new NCTM Standards, as stated in the I-MATH chapter on Symmetry.

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