December 16, 2016
Unit 6, Lesson 3
Converse of the
Pythagorean Theorem
"Every new body of discovery is mathematical in
form, because there is no other guidance we can
have."
-Charles Darwin
In the past we have seen that the converse
of a statement is not always true.
"If I live in Las Vegas, then I live in Nevada."
"If I live in Nevada, then I live in Las Vegas."
December 16, 2016
2
2
2
Example 1: Show if a + b = c , then ΔABC is a right triangle
with the right angle at vertex C.
(Hint: There exists a rightΔDEF
with legs length a and b.)
Converse of the Pythagorean Theorem (Theorem 8-3)
If the square of one side of the triangle is equal to
the sum of the squares of the other two sides,
then the triangle is a right triangle.
Note: When used you can justify by saying "by the Pythagorean Theorem."
December 16, 2016
2
2
2
We now know if a triangle is right if c =a +b ,
2
2
2
so what does it mean if c ≠a +b ?
2
2
2
2
2
2
What type of triangle is it when c <a +b ? c >a +b ?
Explain your thoughts
Recap:
Converse of
Pythagorean Theorem
Theorem 8-4
Theorem 8-5
d
Note: c is the longest side of each triangle.
December 16, 2016
Example 2: A triangle has the side lengths given. Tell whether it
is an acute, obtuse or right triangle. If no triangle can
be formed, then state not a triangle.
a) 12, 8, 10
c) 8, 10, 6
b) 5, 1, 4
d) 2√3, 3, 2√6
December 16, 2016
In Example 2 part c, the sides 6-8-10 are called are a
2
2
2
Pythagorean Triple (3 integers such that a + b = c )
Common Pythagorean Triples:
3, 4, 5
5, 12, 13
8, 15, 17
7, 24, 25
Example 3: Find all values of x so the statement is true.
a)
c)
b)
d) the triangle is isosceles
e) No triangle is possible
December 16, 2016