Pythagorean Theorem and its application to real life situation ppt

Pythagorean Theorem
Right Triangle

1. The Pythagorean The Pythagorean
TheoremTheorem
A A tooltool for for right right
triangletriangle problems problems
onlyonly

2. When do I need to useWhen do I need to use
the Pythagorean the Pythagorean
Theorem ?Theorem ?
When I know: length of 2 sidesWhen I know: length of 2 sides
AndAnd
Need to know: length of 3rd sideNeed to know: length of 3rd side

3. What is the Pythagorean
Theorem?
a
2
b
2
c
2
Where a and b are legs and c is the hypotenuse

4. Using aUsing a
22
+ b + b
22
= c = c
22
Looking for length ofLooking for length of
the hypotenusethe hypotenuse
aa
22
+ b + b
22
= c = c
22
1515
22
+ 20 + 20
22
= c = c
22
225 + 400 = c225 + 400 = c
22
625 = c625 = c
22
625c
2
25c

5. Using aUsing a
22
+ b + b
22
= c = c
22
Looking for length ofLooking for length of
A legA leg
aa
22
+ b + b
22
= c = c
22
66
22
+ b + b
22
= 10 = 10
22
36 + b36 + b
22
= 100 = 100
-36 -36-36 -36
bb
22
= 64 = 64
b = 8b = 8

6. Applying the Applying the
Pythagorean TheoremPythagorean Theorem

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

8. Solving the problemSolving the problem
 aa
22
+ b + b
22
= c = c
22
Write formulaWrite formula
 xx
22
+ 5 + 5
22
= 15 = 15
22
Substitute inSubstitute in
 xx
22
+ 25 = 225 + 25 = 225 Solve for xSolve for x
 - 25 -25- 25 -25
 xx
22
= 200 = 200

 x = 14.142135 x = 14.142135 Check: How am ICheck: How am I
 to leave my answer?to leave my answer?
x200
The ladder reaches 14 feet up the wall.

9. Tim rode 8 miles due north, then 3 miles Tim rode 8 miles due north, then 3 miles
due east. How far, to the nearest mile, is due east. How far, to the nearest mile, is
Tim from where he started? Tim from where he started?
Draw a picture:Draw a picture:

10. Solve the problemSolve the problem
aa
22
+ b + b
22
= c = c
22
88
22
+ 3 + 3
22
= c = c
22
64 + 9 = c64 + 9 = c
22
73 = c73 = c
22
73c
Remember: How
am I suppose to
leave my answer?
C = 8.5440037
Tim is 9 miles from where he started.

11. Further ApplicationsFurther Applications
The diagonals of a rhombus are 6 The diagonals of a rhombus are 6
cm and 8 cm. What is the length cm and 8 cm. What is the length
of each side of the rhombus?of each side of the rhombus?
Prior Knowledge: What properties does a rhombus have?
The diagonals of a rhombus: bisect each other
The diagonals of a rhombus: are perpendicular
The sides of a rhombus: are congruent

12. Draw a picture and solve:Draw a picture and solve:
aa
22
+ b + b
22
= c = c
22
33
22
+ 4 + 4
22
= c = c
22
9 + 16 = c9 + 16 = c
22
25 = c25 = c
22
25c
5 = c
Each side of the rhombus is worth 5 cm.
A rhombus is
really 4 right
triangles in
disguise!

13. As seen in the accompanying diagram, a person As seen in the accompanying diagram, a person
can travel from NYC to Buffalo by going north can travel from NYC to Buffalo by going north
170 miles to Albany and then west 280 miles to 170 miles to Albany and then west 280 miles to
BuffaloBuffalo. .
A) A) If a highway is built to connect NYC and If a highway is built to connect NYC and
Buffalo, how many miles would be saved on the Buffalo, how many miles would be saved on the
trip?trip?
B) With gas prices at $3.10 and a vehicle that B) With gas prices at $3.10 and a vehicle that
gets 18 mpg, how much money would be saved gets 18 mpg, how much money would be saved
roundtrip, if the new highway was traveled roundtrip, if the new highway was traveled
instead of the old route?instead of the old route?

14. Find length of new highwayFind length of new highway
Buf Albany
New York City
170 miles
280 miles
???
a
2
+ b
2
= c
2
280
2
+170
2
=c
2
107300 m= c
2
107300c
2
327.566= c
Did I answer question?
How many miles would be saved?
Old Distance: 280 + 170 = 450Old Distance: 280 + 170 = 450
New Distance: 327.566New Distance: 327.566
Saved Miles: 122.4 or 122 milesSaved Miles: 122.4 or 122 miles

15. How much money can be How much money can be
saved?saved?
Saved Miles: 122 miles x 2 = 244Saved Miles: 122 miles x 2 = 244
Cost to drive one mile (gas):Cost to drive one mile (gas):
$3.10 divided by 18. ($0.1722…)$3.10 divided by 18. ($0.1722…)
Cost to drive 244milesCost to drive 244miles
$ 0.1722 times 244$ 0.1722 times 244
Saved: $42.02Saved: $42.02

16. Wrapping It UPWrapping It UP
The Pythagorean Theorem The Pythagorean Theorem
can be used only on can be used only on
_____triangles._____triangles.
When should the Pythagorean When should the Pythagorean
Theorem be used?Theorem be used?
What should be done first What should be done first
when solving a word when solving a word
problem involving the problem involving the
Pythagorean Theorem?Pythagorean Theorem?
What must be done before What must be done before
writing the answer to a writing the answer to a
Pythagorean Theorem Pythagorean Theorem
problem?problem?
RightRight
When the length of 2 sides are When the length of 2 sides are
known and the length of 3rd side known and the length of 3rd side
is neededis needed
Draw and label triangleDraw and label triangle
Check to see whether the answer Check to see whether the answer
should be rounded or notshould be rounded or not