Transcript Right Triangle Review

Objectives

Use properties of 45° - 45° - 90°
triangles.

Use properties of 30° - 60° - 90°
triangles.
Perfect Squares
The terms of the following sequence:
1, 4, 9, 16, 25, 36, 49, 64, 81…
12,22,32,42, 52 , 62 , 72 , 82 , 92…
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.
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 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 3
5 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 3
5 5
5 3
Practice
Express each square root in its simplest
form by factoring out a perfect square or
by using a factor tree.
12
18
24
32
40
2 3
3 2
2 6
4 2
2 10
48
60
75
83
4 3
2 15
5 3
83
300x3
10 x 3x
Your Turn:
Simplify the expression.
9
64
27
98
10  15
8  28
3 3
7 2
5 6
4 14
3
8
15
4
15
2
11
25
11
5
36
49
6
7
Example 1
Evaluate, and then classify the product.
1.
(√5)(√5) = 5 (rational number)
2.
(2 + √5)(2 – √5) = 451 (rational number)
Conjugates
The radical expressions a + √b and a – √b
are called conjugates.

The product of two conjugates is always
a rational number
Example 2
Identify the conjugate of each of the
following radical expressions:
 7
1. √7
2.
5 – √11 5 11
3.
√13 + 9 9 13
Rationalizing the Denominator
Recall that a radical expression is not in
simplest form if it has a radical in the
denominator. How could we use
conjugates to get rid of any radical in the
denominator and why?
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.
1 3

5 1 3

5  5 3 5  5 3




2
2
1 3 1 3 1 3 1 3
5
Fancy One



Example 3
Simplify the expression.
6
6
6 5
30



5
5
5
5 5
6
7 5

6 7  5

42  6 5


49 5
7 5 7 5




42  6 5 213 5

44
22
17
12
1
9 7

17
17 12


12
12 12
204 2 51
51


12
12
6



1 9  7

 9  7 9  7 
9  7 9  7

81  7
74
Your Turn:
Simplify the expression.
9
8
2
4  11
3 2
4
82 11
5
19
21
399
21
32 4 3
61
8 3
4
Side Lengths of Special Right ∆s

Right triangles whose angle measures
are 45° - 45° - 90° or 30° - 60° - 90° are
called special right triangles. The
theorems that describe the relationships
between the side lengths of each of
these special right triangles are as
follows:
Investigation 1
In this investigation, you will discover a
relationship between the lengths of the
legs and the hypotenuse of an isosceles
right triangle.
This triangle is also referred to as
a 45-45-90 right triangle
because each of its acute
angles measures 45°. Folding
a square in half can make one
of these triangles.
Investigation 1
Find the length of the hypotenuse of each
isosceles right triangle. Simplify the square root
each time to reveal a pattern.
3 2
6 2
4 2
7 2
5 2
12 2
Investigation 1
Did you notice something interesting about
the relationship between the length of
the hypotenuse and the length of the
legs in each problem of this
investigation?
45˚-45˚-90˚ Triangle Theorem
45°-45°-90° Triangle Theorem
In a 45°-45°-90° triangle, the legs ℓ
are congruent and the length of
the hypotenuse ℎ is 2 times the
length of a leg.
Example:
ℓ
ℓ
ℓ
hypotenuse  leg  2
Procedure: Finding the hypotenuse in a
45°-45°-90° Triangle


Find the value of x
By the Triangle Sum
Theorem, the
measure of the third
angle is 45°. The
triangle is a 45°-45°90° right triangle, so
the length x of the
hypotenuse is √2
times the length of a
leg.
3
3
45°
x
Procedure: Finding the hypotenuse in a
45°-45°-90° Triangle
3
3
45°
x
Hypotenuse = √2 ∙ leg
x = √2 ∙ 3
x = 3√2
45°-45°-90° Triangle
Theorem
Substitute values
Simplify
Procedure: Finding a leg in a 45°-45°-90°
Triangle
Find the value of x.
 Because the triangle
is an isosceles right
triangle, its base
angles are
congruent. The
triangle is a 45°-45°90° right triangle, so
the length of the
hypotenuse is √2
times the length x of
a leg.

5
x
x
Procedure: Finding a leg in a
45°-45°-90° Triangle
5
x
Statement:
Reasons:
Hypotenuse = √2 ∙ leg
5 = √2 ∙ x
5
√2
5
√2
√2
√2
x
5
√2
5√2
2
=
√2x
√2
= x
45°-45°-90° Triangle Theorem
Substitute values
Divide each side by √2
Simplify
= x
Multiply numerator and
denominator by √2
= x
Simplify
Example 4a
A. Find x.
The given angles of this triangle are 45° and 90°. This
makes the third angle 45°, since 180 – 45 – 90 = 45.
Thus, the triangle is a 45°-45°-90° triangle.
Example 4a
45°-45°-90° Triangle Theorem
Substitution
Example 4b
B. Find x.
The legs of this right triangle have the same measure,
x, so it is a 45°-45°-90° triangle. Use the
45°-45°-90° Triangle Theorem.
Example 4b
45°-45°-90° Triangle Theorem
Substitution
x = 12
Answer: x = 12
Your Turn:
A. Find x.
A. 3.5
B. 7
C.
D.
Your Turn:
B. Find x.
A.
B.
C. 16
D. 32
Example 5
Find a.
The length of the hypotenuse of a 45°-45°-90° triangle
is
times as long as a leg of the triangle.
45°-45°-90° Triangle Theorem
Substitution
Example 5
Divide each side by
Rationalize the denominator.
Multiply.
Divide.
Your Turn:
Find b.
A.
B. 3
C.
D.
Cartoon Time
Investigation 2
The second special right triangle is the 30˚-60˚-90˚
right triangle, which is half of an equilateral
triangle.
Let’s start by using a little
deductive reasoning to
reveal a useful
relationship in 30˚-60˚-90˚
right triangles.
Investigation 2
Triangle ABC is equilateral, and
segment CD is an altitude.
1. What are m∠A and m∠B?
2. What are m∠ADC and
m∠BDC?
3. What are m∠ACD and
m∠BCD?
4. Is ΔADC ≅ ΔBDC? Why?
5. Is AD=BD? Why?
Investigation 2
Notice that altitude CD divides the
equilateral triangle into two right
triangles with acute angles that measure
30° and 60°. Look at just one of the 30˚60˚-90˚ right triangles. How do AC and
AD compare?
Conjecture:
In a 30°-60°-90° right triangle, if the side
opposite the 30° angle has length s,
then the hypotenuse has length -?-. 2s
Investigation 2
Find the length of the indicated side in each right
triangle by using the conjecture you just made.
17
18
33
10
26
8.5
Investigation 2
Now use the previous conjecture and the
Pythagorean formula to find the length of each
indicated side.
6√3
4√3
5√3
8
6
8
50√3
50
10
4
Investigation 2
You should have notice a pattern in your
answers. Combine your observations
with you latest conjecture and state your
next conjecture.
In a 30˚-60˚-90˚ triangle:
short side = s
hypotenuse = 2s
long side = s√3
30˚-60˚-90˚ Triangle Theorem
30°-60°-90° Triangle Theorem
In a 30°-60°-90° triangle, the length
of the hypotenuse ℎ is 2 times the
length of the shorter leg s, and the
length of the longer leg ℓ is √3
times the length of the shorter leg.
Example:
s
2s
s√3
hypotenuse  2  shorter leg
longer leg  shorter leg  3
Procedure: Finding side lengths in a 30°60°-90° Triangle


Find the values of s
and t.
Because the triangle
is a 30°-60°-90°
triangle, the
longer leg is √3
times the length
s of the shorter
leg.
60°
t
s
30°
5
Procedure: Side lengths in a
30°-60°-90° Triangle
t
30°
Statement:
5 = √3 ∙ s
5
√3
5
√3
√3
s
5
Reasons:
Longer leg = √3 ∙ shorter leg
√3
60°
5
√3
5√3
3
=
√3s
√3
= s
30°-60°-90° Triangle Theorem
Substitute values
Divide each side by √3
Simplify
= s
Multiply numerator and
denominator by √3
= s
Simplify
The length t of the hypotenuse is twice the length s
of the shorter leg.
30°
Statement:
60°
t
s
5
Reasons:
Hypotenuse = 2 ∙ shorter leg
5√3
= 2∙
3
t
t
=
10√3
3
30°-60°-90° Triangle Theorem
Substitute values
Simplify
Example 6
Find x and y.
The acute angles of a right triangle are
complementary, so the measure of the third angle is
90 – 30 or 60. This is a 30°-60°-90° triangle.
Example 6
Find the length of the longer side.
30°-60°-90° Triangle
Theorem
Substitution
Simplify.
Example 6
Find the length of hypotenuse.
30°-60°-90° Triangle
Theorem
Substitution
Simplify.
Answer: x = 4,
Your Turn:
Find BC.
A. 4 in.
B. 8 in.
C.
D. 12 in.
Your Turn:
Shaina designed 2 identical bookends according to
the diagram below. Use special triangles to find the
height of the bookends.
A.
B. 10
C. 5
D.
Two Special Right Triangles
ℓ√2
ℓ
ℓ
2s
s√3
s
45°-45°-90° Special Right Triangle

In a triangle 45°-45°-90° , the hypotenuse is 2 times as long as
a leg.
Example:
45°
45°
Hypotenuse
Leg
ℓ
ℓ
45°
Leg ℓ
5 2cm
5 cm
2
45°
5 cm
30°-60°-90° Special Right Triangle

In a triangle 30°-60°-90° , the hypotenuse is twice as long as the
shorter leg, and the longer leg is 3 times as long as the shorter leg.
Hypotenuse
30°
Example:
2s
Longer Leg
30°
10 cm
s 3
5 3 cm
60°
60°
s
Shorter Leg
5 cm
Find the value of a and b.
7 cm
b = 14
cm
60°
b
a =7 3
cm
30°
2s
s 3
60°
30 °
a
s
Step 1: Find the missing angle measure. 30°
Step 2: Decide which special right triangle applies. 30°-60°-90°
Step 3: Match the 30°-60°-90° pattern with the problem.
Step 4: From the pattern, we know that s = 7 , b = 2s, and a = s 3.
Step 5: Solve for a and b
Find the value of a and b.
7 cm
b=7 2
cm
45°
b
2
45 °
a = 7 cm
a
45°
ℓ
ℓ
2
45°
ℓ
Step 1: Find the missing angle measure. 45°
Step 2: Decide which special right triangle applies. 45°-45°-90°
Step 3: Match the 45°-45°-90° pattern with the problem.
Step 4: From the pattern, we know that ℓ = 7 , a = ℓ, and b = ℓ 2 .
Step 5: Solve for a and b