Color-Coded Flashcards –It is difficult to describe in words which segments to use in the geometric mean to find
the desired segment. Labeling the figure with variables and using a formula is the standard method. The relationship
is easier to remember if the labeling of the triangles is kept the same every time the figure is drawn. What the
students need to remember, is the location of the segments relative to each other. Making color-coded pictures or
flashcards will be helpful. For each relationship the figure should be drawn on both sides of the card. The segment
whose measure is to be found should be highlighted in one color on the front, and on the back, the two segments
that need to be used in the geometric mean should be highlighted with two different colors. Using two colors on the
back is important because the segments often overlap. Making these cards will be helpful even if the students never
use them. Those that have trouble remembering the relationship will use these cards frequently as a reference.
Add a Step and Find the Areas –The exercises in this section have the students find the base or height of triangles.
They have all the information that they need to also calculate the areas of these triangles. Students need practice with
multi-step problems. Having them find the area will help them think through a more complex problem, and give
them practice laying out organized work for calculations that are more complex. Chose to extend the assignment or
not based on how well the students are doing with the material, and how much time there is to work on this section.
Additional Exercise:
- Refer to the figure used to give the relationship of the altitude as the geometric mean of the lengths of the two
segments of the hypotenuse on page 478 of the text.
Letf=3 cm andc=10 cm. What are the values ofdande?
Answer:
3 =
√
e∗( 10 −e) e=1 or 9 cm,sod=9 or1 cm such that the sum is 10
9 = 10 e−e^2
0 =e^2 − 10 e+ 9Special Right Triangles
Memorize These Ratios –There are some prevalent relationships and formulas in mathematics that need to be
committed to long term memory, and the ratios made by the sides of these two special right triangles are definitely
among them. Students will use these relationships not only in the rest of this class, but also in trigonometry, and in
other future math classes. Students are expected to know these relationships, so the sooner learn to use them and
commit them to memory, the better off they will be.
Two is Greater Than the Square Root of Three –One way that students can remember the ratios of the sides of
these special right triangles, is to use the fact that in a triangle, the longest side is opposite the largest angle, and the
shortest side is opposite the smallest angle. At this point in the class, students know that the hypotenuse is the longest
side in a right triangle. What sometimes confuses them is that in the 30− 60 −90 triangle, the ratio of the sides is
1 : 2 :
√
3, and if they do not really think about it, they sometimes put the√
3 as the hypotenuse because it might
seem bigger than 2. Using the opposite relationship is a good method to use when working with these triangles. Just
bring to the students’ attention that 2>
√
3.
TABLE2.13:
45 − 45 −90 Triangle 30 − 60 −90 Triangle
x^2 +x^2 =c^2 x^2 +b^2 = ( 2 x)^2
2 x^2 =c^2 x^2 +b^2 = 4 x^2
x√
2 =c b^2 = 3 x^2Chapter 2. Geometry TE - Common Errors