Transcript Pythagorean Theorem

Aim: What is the Pythagorean Theorem &
how do we use it?
Do Now:
Approximate 32 to the nearest tenth.
Aim: Pythagorean Theorem
Course: Applied Geometry
Square Root
The square root of any real number is a
number, rational or irrational, that when
multiplied by itself will result in a product
that is the original number.
The Radical
Radical
sign
25  5
Square
Root
Radicand
• Every positive number has a positive and negative sq. root.
• The principal Sq. Root of a number is the positive sq. root.
• A rational number can have a rational or irrational sq. rt.
• An irrational number can only have an irrational root.
Aim: Pythagorean Theorem
Course: Applied Geometry
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a right triangle.
Pythagoras.
Pythagorean
Hmm. . .In a right triangle . . .
Cool, huh!
B
c
a
. . the square of the length of the hypotenuse
c is equal to the sum of the squares of the
lengths of the other two sides a and b.
c2 = a2 + b2
C
b
A
52 = 32 + 42
25 = 9 + 16
25 = 25
5
4
3
Aim: Pythagorean Theorem
Course: Applied Geometry
The Square of the What?
F
Pythagorean Theorem
c2 = a2 + b2
A
cc2
2
b2
2
b
c
a
C
B
a2
2
Aim: Pythagorean Theorem
Course: Applied Geometry
Model Problem
A right triangle has sides of lengths 20, 29,
and 21. Which of these is the length of the
hypotenuse?
29
Prove that a triangle with sides 13, 84 and
85 is a right triangle.
Pythagorean Theorem
c2 = a2 + b2
852 = 842 + 132
7225 = 7056 + 169
7225 = 7225
c2 = a2 + b2
Aim: Pythagorean Theorem
Course: Applied Geometry
Model Problem
Find the value of x. Round to nearest tenth.
8
20
x
Pythagorean Theorem
c2 = a2 + b2
202 = 82 + x2
400 = 64 + x2
336 = x2
x = 336
x = 18.3
Aim: Pythagorean Theorem
Course: Applied Geometry
Model Problem
Use the triangle below to find the missing
length. Round to nearest tenth.
a
c
b
Pythagorean Theorem
c2 = a2 + b2
a = 3, b = 7, c = ?7.6
a = ?,
19.3, b = 23, c = 30
a = 1.2, b = ?3.3,, c = 3.5
Aim: Pythagorean Theorem
Course: Applied Geometry
Model Problem
The hypotenuse of a right triangle is 25. If
one leg is 20, the other leg is
1) 5
3) 15
2) 1025
4) 45
Which of the following could be the lengths
of the sides of a right triangle?
1) 3,5,8
2) 5,12,13
3) 2,4,6
4) 5,5,5
Aim: Pythagorean Theorem
Course: Applied Geometry
A ladder is placed 5
feet from the foot of a
wall. The top of the
ladder reaches a point
12 feet above the
ground. Find the
length of the ladder.
c2 = a2 + b2
c = length of ladder = ?
13
b = distance from wall = 5’
a = height above ground = 12’
12’
?
5’
c2 = 122 + 52
c2 = 144 + 25
c2 = 169
c = 13
Cool!
Aim: Pythagorean Theorem
Course: Applied Geometry
A city park department rents paddle boats
at docks near each entrance to the park.
About how far to the nearest meter, is it to
paddle from one dock to the other?
c2 = a2 + b2
dock
c2 = 3502 + 2502
350 m.
a
c2 = 62,500 +
122,500
c
c2 = 185,000
c=
dock
250 m. = b
Aim: Pythagorean Theorem
185,000
c = 430.11626
c = 430 m.
Course: Applied
Geometry
to nearest
meter
Model Problem
A pole
BD, 28 feet high, is perpendicular to the
A pole
ground. Two wires, BC and BA,each 35 feet long,
are attached to the top of the pole and to stakes A
C on the ground. If points A, D, and C are collinear.
how far are the stakes A and C from each other?
B
c2 = a2 + b2
352 = 282 + DC2
28’
35’
A
35’
D
?
C
1225 = 784 + DC2
441 = DC2
21 = DC
AC = 2DC = 2(21) = 42’
Aim: Pythagorean Theorem
Course: Applied Geometry
Model Problem
Find the value of x.
4 5
x = 17.89
?8
4
16
Pythagorean Theorem
c2 = a2 + b2
4 5
2
 4  AB
2
2
x 2  82  162
80 = 16 + AB2
x2 = 320
64 = AB2
x = 17.89
x=8
Aim: Pythagorean Theorem
Course: Applied Geometry
Pythagorean Triplets
Goo
Goo
Goo
a
b
c
3
4
5
5
12
13
8
15
17
For the Pythagorean Theorem,
commonly used numbers that “work
nicely” - and multiples of these
Triplets
There are others. Can you come up with one?
Aim: Pythagorean Theorem
Course: Applied Geometry
Pythagorean Triplets
Goo
Goo
a
Goo
b
c
Find the 3rd side that would make the
following pair a Pythagorean Triplet.
9, 41 and ?
For the Pythagorean Theorem,
commonly used numbers that “work
nicely” - and multiples of these
Triplets
Aim: Pythagorean Theorem
Course: Applied Geometry