Transcript 8.1 Pythagorean Theorem and Its Converse • Pythagorean Theorem – If a triangle is a right triangle, then the sum of the squares of.

8.1 Pythagorean Theorem and Its
Converse
• Pythagorean Theorem
– If a triangle is a right triangle, then the sum of the
squares of the lengths of the legs is equal to the
square of the length of the hypotenuse.
Finding the Length of the Hypotenuse
• What is the length of the hypotenuse in Triangle
ABC?
a b  c
2
2
20  21  c
2
2
2
2
400  441  c
841  c
2
2
841  c
2
29  c
Simplest Radical Form
• Simplify
20.
20  4  5
 4 5
2 5
Simplest Radical Form
• Simplify
192.
192  64  3
 64  3
8 3
Finding the Length of the Leg
• What is the value of x? Express your answer in
simplest radical form.
a b  c
2
2
2
8  x  20
2
2
2
64  x  400
2
x  336
2
x  336
2
x  16(21)
x  4 21
Converse of the Pythagorean Theorem
• If the sum of the squares of the lengths of two sides of
a triangle is equal to the square of the length of the
third side, then the triangle is a right triangle.
Theorems 8.3 and 8.4
• Theorem 8.3 – 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 the triangle is OBTUSE.
• Theorem 8.4 – 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 the triangle is ACUTE.
Pythagorean Triples
• A Pythagorean triple is a set of nonzero whole
numbers, a, b, and c that always satisfy the
equation a² + b² = c².
• Some common Pythagorean Triples:
3, 4, 5
5, 12, 13
8, 15, 17
7, 24, 25
• Keep an eye out for the multiples of these
common Pythagorean Triples (ex. 6, 8, 10)
Classifying a Triangle
• A triangle has side lengths 6, 11, and 14.
Is it acute, obtuse, or right?
c² ___ a² + b²
14² ___ 6² + 11²
196 ___ 36 + 121
196 > 157
The triangle is obtuse.
More Practice!!!!!
• Homework – Textbook p. 495 – 496 #8 –
32 EVEN.