TEXT: Elementary College Geometry by Africk
Geometry Labs by Henri Picciotto. Videos from Khan Academy.
A ratio is just a fancy name for a comparison between 2 quantities, usually with the same units. For instance, the ratio of the mass of the earth to that of the moon is about 80 to 1 and for the diameter it is about 4 to 1. In other words, the earth, 80 times as massive as the moon, will appear 4 times as wide to an outside observer. Click here for more planetary ratio information.
A proportion is an equation stating that 2 ratios or fractions are equal. There are 2 principal tasks with proportions.
- To be in proportion: determine whether the proportion is true, in another words whether 2 ratios are equal. Click here if you are not familiar with this.
- Assume that a proportion is true and solve for unknown:
Often the proportion with an unknown is associated with a word problem:
Here are some additional word problems solved using proportions.
If the proportion is a:b=c:d, then the means are “b” and “c” and the extremes are “a” and “d”. Switching the means or extremes produces equivalent proportions, e.g., a:b=c:d is true if and only if a:c=b:d is true. A common solution technique, cross-multiplication, can be stated: the product of the means is equal to the product of the extremes.
Similar planar figures have the same shape but may be of different sizes. In other contexts, such as architecture or biology, the concept is scale drawings. Polygon similarity means (1) corresponding angles are the same and (2) corresponding sides are in proportion. In general, to determine similarity, it is not enough to have just information about angles or sides. For example, 2 rectangles have the same angles but are not necessarily similar; the ratio between consecutive sides is arbitrary. Rhombi have equal sides, yet may not be similar. Thinking of vertices as hinges, open them up equally to get a square or close them at 2 opposite vertices to get a skinny diamond.
To get similar triangles, it is enough to know that either their angles (AAA) are equal or their sides are in proportion RRR (SSS). In fact, it is enough to know that just 2 sets of angles are equal (AA) since we get the third set of equal angles for free (the 3 angles of a triangle sum to 180°). More surprising, it is enough to know 2 pairs of corresponding sides are in proportion and their included angles are equal RAR (SAS). The first half of this video is a problem requiring RAR (SAS). For a wider exposition of the topic, see the KA video parts I and II; more exercises.
Note: Don’t get confused with our abbreviations. We will encounter SSS and SAS again but in the context of equal or congruent triangles (equal sides as well as angles). Hence, we the instructors have decided to use RRR in place of SSS and RAR in place of SAS (R stands for “ratio”).
Lab 10.1 will give you hands on experience with the geoboard and is designed to cement notions from the day’s material, including scaling and similarity. To do problems 4 and 5, you should be familiar with the notion of slope. Here is a mildly annoying rap video on the subject which you may find humorous.
Check out how a teacher uses a mirror to calculate the heights of students in her class. [On p. 24 of your geometry textbook, the angles of incidence and reflection are defined as in this diagram. In physics, the angles defined are the respective complements. However, the statement “angle of reflection = angle of incidence” is valid for both sets of definitions.]