20-2: Converse of the Pythagorean Theorem
Objectives:
1. To investigate and
use the Converse of
the Pythagorean
Theorem
2. To classify triangles
using the
Pythagorean
formula
Assignment:
• P. 288: 1-11
• P. 290: 11-17, except
16
• Challenge Problems
Objective 1
You will be able to investigate and
use the Converse of the
Pythagorean Theorem
Example 1
1. State the Pythagorean Theorem in if-then
form.
2. What is the Converse of the Pythagorean
Theorem?
Investigation 1
In this investigation you will use string to
construct triangles whose sides have
lengths that are Pythagorean triples. Then
we will see if the triangle you made is
indeed a right triangle.
Investigation 1
Step 1: Select a set of
Pythagorean triples from
the previous list. Mark
off points A, B, C, and D
on a string to create
lengths from your set of
Pythagorean triples.
For example, mark off
segments 8 cm, 15 cm,
and 17 cm long.
Investigation 1
Step 2: Hold the two
ends together (points
A and D) while a
group member holds
point B and another
holds point C.
Carefully stretch the
rope tight to form a
triangle.
Investigation 1
Step 3: With a protractor, measure the
largest angle or use a corner of a piece of
paper to test for a right angle.
Investigation 1
Was your figure close to a right triangle? Try
this investigation again by selecting
another set of lengths from the list of
Pythagorean triples. Do you get a right
triangle?
Theorem!
Converse of the Pythagorean Theorem
If the square of the length of the longest side of a
triangle is equal to the sum of the squares of the
lengths of the other two sides, then it is a right
triangle.
Example 2
Which of the following is a right triangle?
Example 3
The first steel squares
were manufactured by
the Eagle Steel
Company in the
1820s. These are
now used by
carpenters
everywhere to check
and construct right
angles.
Example 3
Prior to the invention of
this tool, carpenters
used a good oldfashioned tape
measure and the
Converse of the
Pythagorean
Theorem. How do
you think this was
accomplished?
Example 4
Tell whether a triangle with the given side
lengths is a right triangle.
1. 5, 6, 7
2. 5, 6, 61
3. 5, 6, 8
Objective 2
You will be able to
classify triangles
using the
Pythagorean
formula
Investigation 2
In the previous
example, the side
lengths 5 and 6 do
not make a right
triangle when paired
with either 7 or 8.
So what kind of
triangle do these
side lengths make?
Theorems!
Acute Triangle Theorem
If the square of the length of the longest side
of a triangle is less than the sum of the
squares of the lengths of the other two
sides, then it is an acute triangle.
Theorems!
Obtuse Triangle Theorem
If the square of the length of the longest side
of a triangle is greater than the sum of the
squares of the lengths of the other two
sides, then it is an obtuse triangle.
Example 5
Can segments with lengths 4.3 feet, 5.2 feet,
and 6.1 feet form a triangle? If so, would
the triangle be acute, right, or obtuse?
Example 6
The sides of an obtuse triangle have lengths
x, x + 3, and 15. What are the possible
values of x if 15 is the longest side of the
triangle?
20-2: Converse of the Pythagorean Theorem
Objectives:
1. To investigate and
use the Converse of
the Pythagorean
Theorem
2. To classify triangles
using the
Pythagorean
formula
Assignment:
• P. 288: 1-11
• P. 290: 11-17, except
16
• Challenge Problems