2.8 Right Triangle Trigonometry
The Pythagorean Theorem
Presenting the Proof –The proof of the Pythagorean theorem given in this lesson provides a wonderful review of,
and use for, what the students just learned about similar triangles. Sometimes it is difficult for students to see the
three right triangles contained in the figure and how the sides correspond. It is helpful to make an additional drawing
of the three triangles so that they are separate, and oriented in the same direction. Using both figures for reference
students can more easily verify the proportions used in this proof.
Skipping Around –Not all texts present material in the same order, and many instructors have a preferred way to
develop concepts that is not always the same as the one used in the text. The Pythagorean theorem is frequently
moved from place to place. If the students have not done similar figures yet, or if area has already been covered,
the proof of the Pythagorean theorem given in the exercises may be the better place to focus the students’ attention.
Proofs are hard for most students to understand. It is important to choose one that the students can feel good about.
Don’t limit the possibilities to these two, research other methods, and pick the one that is most appropriate for your
class. Or better yet, pick the best two or three. Different proofs will appeal to different students.
The Height Must be Measured Along a Segment That is Perpendicular to the Base –When given an isosceles
triangle where the altitude is not explicitly shown, student will frequently try to use the length of one of the sides
of the triangle for the height. The will do this repeatedly, even after you tell them that they must find the length
of the altitude that is perpendicular to the segment that’s length is being used for the base in the formulaA=^12 bh.
Sometimes they do not know what to do, and are just trying something, which is, in a way, admirable. The more
common explanation though is that they forget. The students have been using this formula for years, they think they
know this material, so they just plug and chug, not realizing that the given information has changed. Remind the
students that now that they are in Geometry class, there is an extra step. The new challenge is to find the height, and
then they can do the easy part and plug it into the formula.
Derive the Distance Formula –After doing an example with numbers to show how the distance formula is basically
just the Pythagorean theorem, use variables to derive the distance formula. Most students will understand the proof
if they have seen a number example first. Point out to the students that the number example was inductive reasoning,
and the proof was deductive reasoning. Taking the time to do this is a good review of logic and algebra as well as
great proof practice.
Converse of The Pythagorean Theorem
Mnemonic Devise for Acute and Obtuse Triangles –Many students have trouble remembering that the inequality
with the greater tan is true when the triangle is acute, and that the equation with the less than is true for obtuse
triangles. It seams backwards to them. One way to present this relationship is to compare the longest side and the
angle opposite of it. In a right triangle, the equation has an equal sing; the hypotenuse is the perfect size. When the
longest side of the triangle is shorter than what it would be in a right triangle, the angle opposite that side is also
smaller, and the triangle is acute. When the longest side of the triangle is longer than what is would be in a right
triangle, the angle opposite that side is also larger, and the triangle is obtuse.
Review Operations with Square Roots –Some of the exercises in this section require students to do operations
Chapter 2. Geometry TE - Common Errors