- The document discusses solving a radical equation by isolating the radicals on each side of the equal sign and then squaring both sides. - It provides an example of solving the equation 2x+4=x+7 by squaring both sides, simplifying, and finding the solution x=3. - The final steps are to verify the solution by plugging it back into the original equation.

1. TODAY

2. y = mx + b
Review for Radical Test tomorrow

3. = 2

4. 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 0 Therefore, the
solution is
x = 3.
Solve
Solving Radical Equations
2(3)  4  3  7
10  10
0
L.S. R.S.

5. 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

6. final three...

7. The Pythagorean Theorem

8. 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

9. The Pythagorean Theorem
8"
6"
x"

10. The Pythagorean Theorem

11. The Pythagorean Theorem

12. The Pythagorean Theorem
484 = 81 + 196 ?

13. The Pythagorean Theorem
21"
x"

15. The Pythagorean Theorem
18’9”

16. The Pythagorean Theorem
• Solution: x2 = 902 + 902 = 16,200 x = 127.28 ft

17. Class Work: 4.9
Common
Pythagorean Triples
6 8 10
5 12 13
7 24 25
8 15 17
3 4 5
a2 + b2 = c2
12 16 20

20.  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

21.  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 = c2
a
b
c

22.  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

23.  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.
8
20
x

24. 1. 2.
3. 4.
x
8
6
x
x
x x8
x
26 26
48
2
x
3
3

25. 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

26.  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?

27. 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

28. 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

29. 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

30. The Pythagorean Theorem
Proof of the Pythagorean Theorem:

31. 23 x 12
3(2)2(2)+3(1)2(1)
672
= 276
18 x 37
8(7)
6
24 + 7+ 5
6
3+ 3
6
= 666
Work
Backward
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