Transcript Two Special Right Triangle

Example 1
Write the first 20 terms of the following
sequence:
1, 4, 9, 16, …
x
1
2
3
4
5
6
7
8
9
1
0
1
1
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
2
0
x2
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
256
289
324
361
400
These numbers are called the
Perfect Squares.
Square Roots
The number r is a square root of x if r2 = x.
• This is usually written x  r
• Any positive number has two real square
roots, one positive and one negative, √x
and -√x
√4 = 2 and -2, since 22 = 4 and (-2)2 = 4
• The positive square root is considered the
principal square root
Example 2
Use a calculator to evaluate the following:
1. 3  2
2. 6
3. 3  2
4. 3 / 2
Example 3
Use a calculator to evaluate the following:
1. 3  2
2. 5
3. 3  2
4. 1
Properties of Square Roots
Properties of Square Roots (a, b > 0)
Product Property
ab  a  b
18  9  2  3 2
Quotient Property
a
a

b
b
2
2
2


25
5
25
Simplifying Radicals
Objectives:
1. To simplify square roots
Simplifying Square Root
The properties of square roots allow us to
simplify radical expressions.
A radical expression is in simplest form
when:
1. The radicand has no perfect-square factor
other than 1
2. There’s no radical in the denominator
Simplest Radical Form
Like the number
3/6, 75 is not in
its simplest form.
Also, the
process of
simplification for
both numbers
involves factors.
• Method 1: Factoring
out a perfect square.
75 
25 3 
25  3 
5 3
Simplest Radical Form
In the second
method, pairs of
factors come out
of the radical as
single factors,
but single
factors stay
within the
radical.
• Method 2: Making a
factor tree.
75 
25
5
3
5
5 3
Simplest Radical Form
This method
works because
pairs of factors
are really perfect
squares. So 5·5
is 52, the square
root of which is
5.
• Method 2: Making a
factor tree.
75 
25
5
3
5
5 3
Investigation 1
Express each square root in its simplest form
by factoring out a perfect square or by
using a factor tree.
12
48
18
60
24
75
32
83
40
300x
3
Exercise 4a
Simplify the expression.
27
10  15
9
64
11
25
Exercise 4b
Simplify the expression.
98
8  28
15
4
36
49
Example 5
Evaluate, and then classify the product.
1. (√5)(√5) =
2. (2 + √5)(2 – √5) =
Conjugates are Magic!
The radical expressions a + √b and a – √b
are called conjugates.
• The product of two conjugates is always a
rational number
Example 7
Identify the conjugate of each of the following
radical expressions:
1. √7
2. 5 – √11
3. √13 + 9
Rationalizing the Denominator
We can use conjugates to get rid of radicals
in the denominator:
The process of multiplying the top and
bottom of a radical expression by the
conjugate of the denominator is called
rationalizing the denominator.


5 1 3
5
1 3
5  5 3 5  5 3




2
2
1  3 1 3 1 3 1 3

Fancy One


Exercise 9a
Simplify the expression.
6
5
6
7 5
17
12
1
9 7
Exercise 9b
Simplify the expression.
19
21
9
8
2
4  11
4
8 3
Solving Quadratics
If a quadratic equation has no linear term,
you can use square roots to solve it.
By definition, if x2 = c, then x = √c and
x = −√c, usually written x = ±√c
– You would only solve a quadratic by finding a
square root if it is of the form
ax2 = c
– In this lesson, c > 0, but that does not have to
be true.
Solving Quadratics
If a quadratic equation has no linear term,
you can use square roots to solve it.
By definition, if x2 = c, then x = √c and
x = -√c, usually written x = √c
– To solve a quadratic equation using square
roots:
1. Isolate the squared term
2. Take the square root of both sides
Exercise 10a
Solve 2x2 – 15 = 35 for x.
Exercise 10b
Solve for x.
1
2
 x  4   11
3
The Quadratic Formula
Let a, b, and c be real numbers, with a ≠ 0.
The solutions to the quadratic equation
ax2 + bx + c = 0 are
b  b  4ac
x
2a
2
Exercise 11a
Solve using the quadratic formula.
x2 – 5x = 7
Exercise 11b
Solve using the quadratic formula.
1. x2 = 6x – 4
2. 4x2 – 10x = 2x – 9
3. 7x – 5x2 – 4 = 2x + 3
The Discriminant
Discriminant
In the quadratic formula, the expression
b2 – 4ac is called the discriminant.
Converse of the Pythagorean Theorem
Objectives:
1. To investigate and use the Converse of
the Pythagorean Theorem
2. To classify triangles when the
Pythagorean formula is not satisfied
Theorem!
Converse of the Pythagorean Theorem
If the square of the length of the longest side
of a triangle is equal to the sum of the
squares of the lengths of the other two
sides, then it is a right triangle.
Example
Which of the following is a right triangle?
Example
Tell whether a triangle with the given side
lengths is a right triangle.
1. 5, 6, 7
2. 5, 6, 61
3. 5, 6, 8
Theorems!
Acute Triangle Theorem
If the square of the length of the longest side
of a triangle is less than the sum of the
squares of the lengths of the other two
sides, then it is an acute triangle.
Theorems!
Obtuse Triangle Theorem
If the square of the length of the longest side
of a triangle is greater than the sum of the
squares of the lengths of the other two
sides, then it is an obtuse triangle.
Example
Can segments with lengths 4.3 feet, 5.2 feet,
and 6.1 feet form a triangle? If so, would
the triangle be acute, right, or obtuse?
Example 7
The sides of an obtuse triangle have lengths
x, x + 3, and 15. What are the possible
values of x if 15 is the longest side of the
triangle?