1. The Pythagorean Theorem and Its Converse

2. History of the Theorem Pythagoras of Samos was a Greek
philosopher responsible for many important
developments in mathematics!
But, rumour has it Pythagoras’ Theorem was known to the Babylonians some 1000 years
before Pythagoras.
However, we all believe he was the
first person to prove the theorem and
that is why the theorem takes his name.

3. Parts of a Right Triangle
In a right triangle, the side opposite the right angle is called the hypotenuse.
It is the longest side.
The other two sides are called the legs.

4. The Pythagorean Theorem
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.
a2 + b2 = c2

5. Pythagorean Triples
A Pythagorean triple is a set of nonzero whole numbers that satisfy the Pythagorean Theorem.
Some common Pythagorean triples include:
3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25
If you multiply each number in the triple by the same whole number, the result is another Pythagorean triple!

6. Pythagoras Questions 1 2 x 3 cm 4 cm Pythagorean triple x 5 cm 12 cm
Pythagorean triple
5 cm
12 cm x 2
Pythagorean triple

7. x ≈ 6.32 cm x ≈ 21.11 cm Pythagoras Questions: Finding a leg measure 3
x m
9 m
11m 3
x ≈ 6.32 cm
Another method for finding a leg measure
11 cm
x cm
23.8 cm 4
x ≈ cm

8. Applications of Pythagoras
Find the diagonal of the rectangle
6 cm
9.3 cm 1 d
d = cm

9. x ≈ 6.51 cm Perimeter = 2(6.51+4.3) ≈ 21.62 cm 2
A rectangle has a width of 4.3 cm and a diagonal of 7.8 cm.
Find its perimeter. 2
7.8 cm
4.3 cm
x cm
x ≈ 6.51 cm
Perimeter = 2( ) ≈ cm
therefore

10. Finding the Length of the Hypotenuse
What is the length of the hypotenuse of ABC? Do the sides form a Pythagorean triple?

11.  The legs of a right triangle have lengths 10 and 24
 The legs of a right triangle have lengths 10 and 24. What is the length of the hypotenuse? Do the sides form a Pythagorean triple?

12. Finding the Length of a Leg
What is the value of x? Express your answer in simplest radical form.

13.  The hypotenuse of a right triangle has length 12
 The hypotenuse of a right triangle has length 12. One leg has length 6. What is the length of the other leg? Express your answer in simplest radical form.

14. Triangle Classifications
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 the triangle is a right triangle.
If c2 = a2 + b2, than ABC is a right triangle.

15. The converse of the Pythagorean Theorem can be used to categorize triangles.
If c2 = a2 + b2, then triangle ABC is a right triangle.
If c2 > a2 + b2, then triangle ABC is an obtuse triangle.
If c2 < a2 + b2, then triangle ABC is an acute triangle.

16. Triangle Inequality The triangle is obtuse 38, 77, 86 c2 ? a2 + b2
862 ?
7396 ?
7396 > 7373
The triangle is obtuse

17. Triangle Inequality The triangle is acute 10.5, 36.5, 37.5
c2 ? a2 + b2
37.52 ? ?
<
The triangle is acute

18. 4,7,9
9²__4² + 7²
81__
81 > 65
OBTUSE
greater

19. 5,5,7
7² __5² + 5² __
49 < 50
ACUTE
Less than