1. Dr. Fowler – CCM1A Unit 1 – Lesson 9 Pythagorean Theorem

2. Warmup Solve for x x2+7=43 Ans: x = ±6 64+x2=164
Evaluate for a = 12, b = 5, c = 13
a2 + b2
c2 – b2
Ans: x = ±6
Ans: x = ±10
Ans: 169
Ans: 144

3. 550 B.C.

4. Pythagoras Pythagoras was a Greek philosopher and religious leader.
He was responsible for many important developments in maths,
astronomy,
and music.

5. The Secret Brotherhood
His students formed a secret society called the Pythagoreans.
As well as studying math, they were a political and religious organization.
Members could be identified by a five pointed star they wore on their clothes.

6. The Secret Brotherhood
They had to follow some unusual rules. They were not allowed to wear wool, drink wine or pick up anything they had dropped!
Eating beans was also strictly forbidden!

7. Here we have a triangle with the lengths of each of the three sides5 4 3

8. Let’s take the lengths of each side and make a square for each of them5 4 3

9. Let’s find the area of each square?

10. Now, let’s add the two smaller areas together.25 16 + 9

11. It turns out this is true for every right triangle
Notice how the sum of the two smaller squares equals the larger square? 25
It turns out this is true for every right triangle +
= ? 9 16

12. The Pythagorean Theorem states: “The sum of the squares of the legs of a right triangle are equal to the square of the hypotenuse.” +
= 25 9 16

13. The Pythagorean Theorem
a2 + b2 = c2 c a b
Where a and b are legs and c is the hypotenuse
The Hypotenuse is always the BIGGEST value

14. Pythagorean Theorem hypotenuse leg leg
The hypotenuse is always the longest side of a right triangle and is always opposite the right angle.

15. Pythagorean Theorem
What is the value of the missing side? 5 12

16. Pythagorean Theorem
What is the value of the missing side? 15 9

17. Using a2 + b2 = c2 a2 + b2 = c2 152 + 202 = c2 225 + 400 = c2 625 = c2
Looking for length of
the hypotenuse
a2 + b2 = c2
= c2
= c2
625 = c2 x 15 20

18. Using a2 + b2 = c2 a2 + b2 = c2 62 + b2 = 102 36 + b2 = 100 -36 -36 10
Find the Leg
a2 + b2 = c2
62 + b2 = 102
36 + b2 = 100
b2 = 64
b = 8 10 6 x

19. Example 1. Find the length of AC.
Hypotenuse 16 B C 12
Solution :
AC2 = (Pythagoras’ Theorem)
AC2 =
AC2 = 400
AC = 20

20. d Example 2. Find the length of diagonal d . Solution:10 24 d
Solution:
d2 = (Pythagoras’ Theorem)

21. Applying the Pythagorean Theorem

22. Application of Pythagoras’ Theorem
1.A car travels 16 km from east to west. Then it turns left and travels a further 12 km. Find the displacement between the starting point and the destination point of the car.
16km N
12km ?

23. Worksheet?

24. What is Volume? Volume is the measure of the capacity of a container.
It is the measure of how much a container of a particular shape will hold - liquids, dry substances, etc.

25. AC2 = AB2 + BC2 (Pythagoras’ Theorem) AC2 = 162 + 122 AC2 = 400
16 km
12 km A B C
Solution :
In the figure,
AB = 16
BC = 12
AC2 = AB2 + BC2 (Pythagoras’ Theorem)
AC2 =
AC2 = 400
AC = 20
The displacement between the starting point and the destination point of the car is 20 km

26. 2. The height of a tree is 5 m. The distance between
the top of it and the tip of its shadow is 13 m.
Find the length of the shadow L.
Solution:
132 = 52 + L2 (Pythagoras’ Theorem)
L2 =
L2 = 144
L = 12
5 m
13 m L

27. A 15 foot ladder leans up against a building
A 15 foot ladder leans up against a building. The foot of the ladder is 5 feet from the base of the building. How high up the wall, to the nearest foot does the ladder reach?
Draw a picture: 15 x 5

28. Solving the problem The ladder reaches 14 feet up the wall.
a2 + b2 = c2 Write formula
x = 152 Substitute in
x = 225 Solve for x
x2 = 200
x = Check: How am I
to leave my answer?
The ladder reaches 14 feet up the wall.

29. Pythagoras Theorem H B L
A boat sails due East from a Harbour (H), to a marker buoy (B),15 miles away. At B the boat turns due South and sails for 6.4 miles to a Lighthouse (L). It then returns to harbour. What is the total distance travelled by the boat? H B L
15 miles
6.4 miles
Total distance travelled = = 37.7 miles

30. Pythagoras Theorem
A 12 ft ladder rests against the side of a house. The top of the ladder is 9.5 ft from the floor. How far is the base of the ladder from the house?
12 ft
9.5 ft L

31. You try this one in your notes.
TAKE
NOTES
You try this one in your notes. x
Find x
5 ft
12 ft
= x2
= x2
169 = x2
Answer:
x = 13

32. Try this one in your notes…x 15 20
Solve for x.
Round your answer to the nearest hundredth if necessary.
Answer: 25

33. Try this one in your notes…7 12 x
Solve for x.
Round your answer to the nearest hundredth if necessary.
Answer:
13.89

34. Try this one in your notes…5 x 3
Solve for x.
Round your answer to the nearest hundredth if necessary.
Answer: 4