Course: Applied Geometry Aim: Pythagorean Theorem Aim: What is the Pythagorean Theorem & how do we...
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Course: Applied Geometry Aim: Pythagorean Theorem Aim: What is the Pythagorean Theorem & how do we use it? A pproxim ate 32 to the nearesttenth. Do Now:
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Transcript of Course: Applied Geometry Aim: Pythagorean Theorem Aim: What is the Pythagorean Theorem & how do we...
Course: Applied GeometryAim: Pythagorean Theorem
Aim: What is the Pythagorean Theorem & how do we use it?
Approximate 32 to the nearest tenth.
Do Now:
Course: Applied GeometryAim: Pythagorean Theorem
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.
525 The RadicalThe Radical
Square Root
Radicand
Radical sign
• 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.
Course: Applied GeometryAim: Pythagorean Theorem
Yasoo, my name is
Pythagoras.
. . 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.
Since I’ve had a lot of free time on my hands, I thought I’d look at some properties of a right triangle. Hmm. . .In a right triangle . . .
4
3
5
a
b
c
C A
B
c2 = a2 + b2
52 = 32 + 42
25 = 9 + 1625 = 25
It’s so important, they named it after me: The Pythagorean Theorem.
Cool, huh!
Course: Applied GeometryAim: Pythagorean Theorem
The Square of the What?
a
b c
C B
A
c2
F
b2
a2
H
c2
b2
a2
c2 = a2 + b2
Pythagorean Theorem
Course: Applied GeometryAim: Pythagorean Theorem
A right triangle has sides of lengths 20, 29, and 21. Which of these is the length of the hypotenuse?
Prove that a triangle with sides 13, 84 and 85 is a right triangle.
852 = 842 + 132
7225 = 7056 + 169
7225 = 7225
29
Model Problem
c2 = a2 + b2
c2 = a2 + b2
Pythagorean Theorem
Course: Applied GeometryAim: Pythagorean Theorem
Find the value of x. Round to nearest tenth.
208
x
202 = 82 + x2
400 = 64 + x2
336 = x2
x = 336
x = 18.3
Model Problem
c2 = a2 + b2
Pythagorean Theorem
Course: Applied GeometryAim: Pythagorean Theorem
Use the triangle below to find the missing length. Round to nearest tenth.
ca
b
Model Problem
a = 3, b = 7, c = ?
a = ?, b = 23, c = 30
c2 = a2 + b2
Pythagorean Theorem
a = 1.2, b = ? , c = 3.5
7.6
19.3,
3.3,
Course: Applied GeometryAim: Pythagorean Theorem
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 3) 2,4,6
2) 5,12,13 4) 5,5,5
Course: Applied GeometryAim: Pythagorean Theorem
c = length of ladder = ? b = distance from wall = 5’ a = height above ground = 12’
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.
5’
12’?
c2 = a2 + b2
c2 = 122 + 52
c2 = 144 + 25
c2 = 169
c = 13Cool!
13
Course: Applied GeometryAim: Pythagorean Theorem
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?
350 m.
250 m.
c2 = a2 + b2
c2 = 3502 + 2502
c2 = 62,500 + 122,500
c2 = 185,000
c = 000,185
c = 430.11626
c = 430 m. to nearest meter
ca
= b
dock
dock
Course: Applied GeometryAim: Pythagorean Theorem
c2 = a2 + b2
352 = 282 + DC2
1225 = 784 + DC2
441 = DC2
Model Problem
A pole A pole , 28 feet high, is perpendicular to the
ground. Two wires, and ,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
BD
BC BA
.
how far are the stakes A and C from each other?B
D
28’
CA
35’35’
?21 = DC
AC = 2DC = 2(21) = 42’
Course: Applied GeometryAim: Pythagorean Theorem
Model Problem
Find the value of x.
4 16
x4 5 ?
80 = 16 + AB2
64 = AB2
x = 8
2 224 5 4 AB
c2 = a2 + b2
Pythagorean Theorem
x = 17.89
2 2 28 16x
x2 = 320
8= 17.89
Course: Applied GeometryAim: Pythagorean Theorem
Pythagorean TripletsPythagorean Triplets
GooGooGoo
3 4 5 3 4 5
5 12 13 5 12 13
8 15 17 8 15 17
For the Pythagorean Theorem, commonly used numbers that “work
nicely” - and multiples of these TripletsTriplets
For the Pythagorean Theorem, commonly used numbers that “work
nicely” - and multiples of these TripletsTriplets
a b c a b c
There are others. Can you come up with one?
Course: Applied GeometryAim: Pythagorean Theorem
Pythagorean TripletsPythagorean Triplets
GooGooGoo
For the Pythagorean Theorem, commonly used numbers that “work
nicely” - and multiples of these TripletsTriplets
For the Pythagorean Theorem, commonly used numbers that “work
nicely” - and multiples of these TripletsTriplets
a b c a b c
Find the 3rd side that would make the following pair a Pythagorean Triplet.
9, 41 and ?
