may 11, 2015
TRANSCRIPT
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TODAY
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y = mx + b
Review for Radical Test tomorrow
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= 2
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2x 4 x 7 0.
Set up the equation so that
there will be one radical on
each side of the equal sign.
2x 4 x 7
Square both sides.
2x 4 2
x 7 2
Simplify.
2x + 4 = x + 7
x = 3
Verify your solution.
2x 4 x 7 0Therefore, the
solution is
x = 3.
Solve
Solving Radical Equations
2(3) 4 3 7
10 10
0
L.S. R.S.
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0)105()23( xx
)105()23( xx
22
)105()23( xx
10523 xx
1022 x
x212
Isolate the radicals
Square both sides
Simplify
Subtract 3x from both sides
Add 10 to both sides
When there are two radicals on the same side of an equation, isolate both radicals by moving one to the other side of the equal sign.
x = 6
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final three...
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The Pythagorean Theorem
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Pythagoras of Samos
• Lived in southern Italy from 571 BC-495 BC
• Considered the first true mathematician
• Used mathematics as a means to understand the natural world
• First to teach that the earth was a sphere that revolves around the sun
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The Pythagorean Theorem
8"
6"
x"
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The Pythagorean Theorem
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The Pythagorean Theorem
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The Pythagorean Theorem
484 = 81 + 196 ?
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The Pythagorean Theorem
21"
x"
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The Pythagorean Theorem
18’9”
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The Pythagorean Theorem
• Solution: x2 = 902 + 902 = 16,200 x = 127.28 ft
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Class Work: 4.9
Common Pythagorean Triples
6 8 10
5 12 137 24 25
8 15 17
3 4 5
a2 + b2 = c2
12 16 20
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In a right triangle, the side opposite the right angle is the longest side. It is the hypotenuse. The other two sides are the legs of a right triangle.
legs
hypotenuse
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In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.
a2 + b2 = c2a
b
c
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A right triangle has sides of lengths 20, 29,
and 21. What is the length of the
hypotenuse?
Verify that the Pythagorean Theorem is true
for the right triangle in the previous
question.
Find the length of the hypotenuse of a right
triangle with legs of lengths 7 and 24.
WHO? Czech-American mathematician
Olga Taussky-Todd (1906-1995) studied
Pythagorean triangles. In 1970, she won
the Ford Prize for her research
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Find the value of x. Leave your answer in
simplest radical form.
The hypotenuse of a right triangle has length
12. one leg has length 6. Find the length of
the other leg in simplest radical form.
820
x
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1. 2.
3. 4.
x
8
6
x
x
x x8
x26 26
48
2x
3
3
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When the lengths of the sides of a right
triangle are integers, the integers form a
Pythagorean Theorem. Here are some
common primitive Pythagorean Triples.
3, 4, 5
5, 12, 13
8, 15, 17
7, 24, 25
Choose an integer. Multiply each number of
a Pythagorean triple by that integer. Verify
that the result is a Pythagorean triple.
9, 40, 41
11, 60, 61
12, 35, 37
13, 84, 85
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What is the length of the diagonal of a rectangle whose sides measures 5 and 7?
Calculate the length of the side of a square whose diagonal measures 9 cm.
What is the measure of the longest stick we can put inside a 3 cm x 4 cm x 5 cm box?
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In ΔABC with longest side c,
if c2 = a2 + b2, then the triangle is right.
if c2 > a2 + b2, then the triangle is obtuse.
if c2 < a2 + b2, then the triangle is acute.
B
C A
a
b
c
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a) 2, 3, 4
b) 3, 4, 5
c) 4, 5, 6
d) 3, 3, 3 2
e) 3, 3, 3 3
f) 2, 2 3, 4
g) 5, 5, 5
h) 4, 4, 5
i) 2, 2, 2
j) 2.5, 6, 6.5
The number represent the lengths of the sides of a triangle (a, b, c). Classify each triangle as acute, obtuse, or right.
obtuse
right
acute
right
obtuse
right
acute/equi
acute
right
right
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a) 2, 3, 4
b) 3, 4, 5
c) 4, 5, 6
d) 3, 3, 3 2
e) 3, 3, 3 3
f) 2, 2 3, 4
g) 5, 5, 5
h) 4, 4, 5
i) 2, 2, 2
j) 2.5, 6, 6.5
The number represent the lengths of the sides of a triangle (a, b, c). Classify each triangle as acute, obtuse, or right.
obtuse
right
acute
right
obtuse
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The Pythagorean Theorem
Proof of the Pythagorean Theorem:
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23 x 123(2)2(2)+3(1)2(1)
672= 276
18 x 378(7)
624 + 7+ 5
6 3+ 3
6= 666
WorkBackward
Variables represent numbers that are unknown at the time. Even
after the variables are known, all of the rules of algebra still apply.
For example, let's foil these two terms:
60"36"
x