SUBJECT: Geometry
TOPIC: Applications of the Pythagorean Theorem and Special
Right Triangles
THEME:
The focus of this lesson will be on applications of the Pythagorean Theorem to word problems
and the topic of special right triangles. The prior knowledge that I will assume for my students is
algebra topics such as taking the square root of a number. I will also assume they know and
understand the Pythagorean Theorem from the previous day's lesson. By learning the ratios of the
sides for some special angle triangles, namely the 45-45-90 and 30-60-90 triangles, it can then be
used to solve problems involving special right triangles. The purpose of this lesson will be on
ensuring an understanding of the concepts well enough to be able to apply the concepts to word
problems. Furthermore, this lesson will prepare the students to make connections in our next
lesson between these concepts and real life situations.
Note to teachers: This lesson is meant to be taught in one 50 minute
class period, but you may have to make some modifications in order
to accomplish all of this in one class meeting. For example, you
might want to work fewer sample problems.
OBJECTIVES:
1. Students should learn to apply the Pythagorean Theorem to
word problems.
2. Students should be able to determine the lengths of two
sides of a 45-45-90 or 30-60-90 triangle when the length of the
third side is given.
Note to teachers: Below are the procedures for the lesson. Whenever it says work example problems, you can find these problems and their solutions by going to the link at the bottom of this page for the class notes. There is a link to the GSP activites at the bottom of this page as well.
PROCEDURE:
1. Briefly review the previous day's lesson on the Pythagorean
Theorem and ask for any questions concerning homework. (Work problems
on board when needed)
2. Work example problems 1 and 2 that apply the Pythagorean
Theorem to word problems on the overhead.
3. Students will complete a GSP activity that will allow them to discover the 45-45-90 triangle theorem and the 30-60-90 triangle theorem.
4. State, explain, and prove the 45-45-90 triangle theorem
and 30-60-90 triangle theorem.
5. Work example problems 3 and 4 that involve 45-45-90 triangles
on an overhead.
6. Work example problems 5 and 6 that involve 30-60-90 triangles
on an overhead.
7. Teacher will call on students to apply the theorem to the
provided examples.
Note to teachers: Below are some suggested questions to ask your students during the lesson. These should be used to encourage students to pay attention and to keep them actively involved in the lesson.
QUESTIONS TO ASK STUDENTS:
1. Ask a student to state the Pythagorean Theorem when reviewing
the previous day's lesson.
2. Ask students how they would solve the word problems that
apply the Pythagorean Theorem.
3. Ask the students what they discovered from the GSP activity.
Call on students to tell the class what they discovered.
4. When working example problems 3 and 4, ask a student what
the ratio of the hypotenuse to the leg is for a 45-45-90 triangle.
5. When working example problems 5 and 6, ask a student to
state how you find the hypotenuse and the longer leg of a 30-60-90
triangle when you know the length of the shorter leg.
EVALUATION:
Students will be required to complete worksheets for homework consisting of problems involving applications of the Pythagorean Theorem and special right triangles. The students should be given about 10 minutes at the end of class to start their homework so that they can get any questions answered that they may have.
Note to teachers: You can find links to these worksheets at the bottom of this page.
MATERIALS:
-overhead projector and pens
-computers with Geometer's Sketchpad software
-worksheets for all of the students to complete for homework
Note to teachers: Below are the GSP sketches I mentioned in the procedures. The triangles are already constructed in order to save time. However, if you have extra time you may want to have the students actually construct the triangles. The questions to go along with the sketches are listed below.
GSP activity-45-45-90 triangles sketch
Activity:
1. Measure the angles of the triangle. How can you classify the triangle by its angles?
2. Measure the lengths of the two legs of the triangle. What do you notice about these lengths? Move the points on the triangle around and see if your conjecture always works.
3. How can you classify the triangle by its sides?
4. What is the relationship between the legs and the hypotenuse?
5. Is this relationship always true? Move the points on the triangle around to test your conjecture.
GSP activity-30-60-90 triangle sketch
Activity:
1. Measure the angles of the triangle. How can you classify the triangle by its angles?
2. Measure the lengths of the sides of the triangle?
3. What could you do to the short leg to get the length of the hypotenuse? What could you do to the long leg to get the length of the hypotenuse?
4. Move the points around on the triangle. Does your conjecture always work?