1. Pythagorean Theorem

2. Right Triangle Hypotenuse Legs Pythagorean Theorem Radical/Radicand
VOCABULARY
Right Triangle
Hypotenuse
Legs
Pythagorean Theorem
Radical/Radicand
Square Root

3. RIGHT TRIANGLE
Longest side is the hypotenuse, side c (opposite the 90o angle)
The other two sides are the legs, sides a and b
Pythagoras developed a formula for finding the length of the sides of any right triangle

4. RADICAL/RADICAND

5. SQUARE ROOT
Definition: A number the produces a specified quantity when multiplied by itself.

6. THE PYTHAGOREAN THEOREM
“For any right triangle, the sum of the areas of the two small squares is equal to the area of the larger.”
a2 + b2 = c2
In different words…
If the angle opposite the hypotenuse is a right angle, then a2 + b2 = c2

7. EXAMPLE #1
If two measures of the sides of a right triangle are known, the Pythagorean Theorem can be used to find the measure of the third side.

8. EXAMPLE #2
152+82=c2
225+64=c2
289=c2
17=c

9. YOUR TURN! ☺

10. Normal: If x, then y. Converse: If y, then x. CONVERSE? Example:
If it is raining, then the grass is wet.
Converse: If the grass is wet, then it is raining.

11. CONVERSE OF THE PYTHAGOREAN THEOREM
“If a triangle has sides of lengths a, b, and c; and a2 + b2 = c2
then the triangle is a right triangle with hypotenuse of length c.”
You can determine if a triangle is a right triangle if
the 3 side lengths fit into a2 + b2 = c2

12. No! This is not a right triangle.
EXAMPLE #1
No! This is not a right triangle.
52+72=92
25+49=81?

13. YOUR TURN! ☺

14. COMMON PYTHAGOREAN TRIPLES

15. Find the length of the hypotenuse if 1. a = 12 and b = 16.
= c2
= c2
400 = c2
Take the square root of both sides.
20 = c

16. Find the length of the hypotenuse if 2. a = 5 and b = 7.16

17. Find the length of the hypotenuse given a = 6 and b = 12
180
324
13.42 18

18. Find the length of the leg, to the. nearest hundredth, if 3
Find the length of the leg, to the nearest hundredth, if 3. a = 4 and c = 10.

19. Find the length of the leg, to the nearest hundredth, if 4
Find the length of the leg, to the nearest hundredth, if 4. c = 10 and b = 7.

20. Find the length of the missing side given a = 4 and c = 51 3
6.4 9

21. 5. The measures of three sides of a triangle are given below
5. The measures of three sides of a triangle are given below. Determine whether each triangle is a right triangle , 3, and 8

22. The rectangular bottom of a box has an interior length of 19 inches and an interior width of 14 inches. A stick is placed in the box along the diagonal of the bottom of the box. Which measurement is closest to the longest possible length for the stick?
a2 + b2 = c2
F 12.8 in.
G 16.3 in.
H 23.6 in.
J 33.0 in.

23. REAL-LIFE APPLICATIONS
The Pythagorean theorem has far-reaching ramifications in other fields (such as the arts), as well as practical applications.
The theorem is invaluable when computing distances between two points, such as in navigation and land surveying.
Another important application is in the design of ramps. Ramp designs for handicap-accessible sites and for skateboard parks are very much in demand.
Can you think of other ways the Pythagorean Theorem can be extremely valuable?

24. Baseball Problem A baseball “diamond” is really a square.
You can use the Pythagorean theorem to find distances around a baseball diamond.

25. BASEBALL PROBLEM The distance between consecutive bases is 90
feet. How far does a
catcher have to throw
the ball from home
plate to second base? 25

26. BASEBALL PROBLEM
To use the Pythagorean theorem to solve for x, find the right angle.
Which side is the hypotenuse?
Which sides are the legs?
Now use: a2 + b2 = c2 26

27. BASEBALL PROBLEM SOLUTION
The hypotenuse is the distance from home to second, or side x in the picture.
The legs are from home to first and from first to second.
Solution:
x2 = = 16,200
x = ft 27

28. LADDER PROBLEM
A ladder leans against a second-story window of a house. If the ladder is 25 meters long, and the base of the ladder is 7 meters from the house, how high is the window? 28

29. LADDER PROBLEM SOLUTION
First draw a diagram that shows the sides of the right triangle.
Label the sides:
Ladder is 25 m
Distance from house is 7 m
Use a2 + b2 = c2 to solve for the missing side.
Distance from house: 7 meters 29

30. LADDER PROBLEM SOLUTION
b = 24 m
How did you do? B 30

31. DO NOT DO THIS! 31