Pythagorean Theorem, Triples, and Inequalities Video Notes
Recall: The Pythagorean Theorem states that in a right triangle, the sum of the squares of the lengths of the legs equals
the square of the length of the hypotenuse.
A
B
C
A set of three nonzero whole numbers a , b , and c such that a 2  b2  c 2 is called a Pythagorean Triple.
Examples of Common Triples
Example 1) Use Pythagorean triples to find the value of 𝑧.
Justify your answer.
13
a) 3, 4, 5
4
b) 5, 12, 13
12
(5z-2)
c) 8, 15, 17
d) 7, 24, 25
Pythagorean Inequalities Theorem:
Given 3 sides lengths that form a triangle, if a 2  b2  c 2 , then you can conclude the 3 side lengths are part of a right
triangle. However, if 𝑎2 + 𝑏 2 ≠ 𝑐 2 , you can follow the following rules:
In ABC , c is the longest side.
If c2  a 2  b2 , then ABC is an obtuse triangle.
If c2  a 2  b2 , then ABC is an acute triangle.
A
A
c
C
c
b
b
a
B
C
a
B
Example 2) The given lengths below form a triangle. Use the theorem above to classify the triangle as acute, obtuse, or
right.
a) 8, 11, and 13
c) 7, 12, and 16
b) 6, 3, and 3 3
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Pythagorean Theorem, Triples, and Inequalities Practice
For problems 1 – 3, find the value of 𝒙. Give your answer in simplest radical form.
1)
2)
3) You will need to FOIL on this one!
For problems 4 – 6, determine which side lengths below form a Pythagorean Triple. Justify your answer by finding the
length of the missing side.
4)
5)
6)
The given lengths in problem 7 – 9 form a triangle. Use the Pythagorean Inequalities Theorem to classify the triangle
as acute, obtuse, or right.
7) 15, 18, 20
8) 7, 8, 11
9) 6, 7, 3√13