Algebraic Roots and Radicals
Simplifying Perfect Square Radical Expressions
Approximating Square Roots
Rational & Irrational Numbers
Radical Expressions Containing Variables
Simplifying Non-Perfect Square Radicands
Simplifying Roots of Variables
Operations with Radicals
Pythagorean Theorem
Distance Formula
Intro to Trig
Solving Right Triangles
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to go to that
section.
Simplifying Perfect Square
Expressions
R
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Table of
Contents
Can you recall the perfect squares from 1 to 169?
12 =
82 =
22 = 92 =
32 = 102 =
42 = 112 =
52 = 122 =
62 = 132 =
202 =
72 =
Square Root Of A Number
Recall: If b2 = a, then b is a square root of a.
Example: If 42 = 16, then 4 is a square root of 16
What is a square root of 25? 64? 100?
Square Root Of A Number
Square roots are written with a radical symbol
Positive square root:
Negative square root: -
=4
= -4
Positive & negative square roots:
=
4
Negative numbers have no real square roots
no real roots because there is no real number that, when squared, would equal
-16.
Is there a difference between
&
Which expression has no real roots?
Evaluate the expression
?
Evaluate the expression
is not real
1
2
?
3
=?
4
5
6
=?
A
3
B
-3
C
No real roots
7
The expression equal to
is equivalent to a positive integer when b is
A
-10
B
64
C
16
D
4
Square Roots of Fractions
a
=
b
16
49
=
b 0
=
4
7
Try These
8
A
C
B
D
no real solution
9
A
C
B
D
no real solution
10
A
C
B
D
no real solution
11
A
C
B
D
no real solution
12
A
C
B
D
no real solution
Square Roots of Decimals
Recall:
To find the square root of a decimal, convert the decimal to a fraction first.
Follow your steps for square roots of fractions.
= .2
= .05
= .3
13
Evaluate
A
B
C
D
No Real Solution
14
Evaluate
A
.06
B
.6
C
6
D
No Real Solution
15
Evaluate
A
.11
B
11
C
1.1
D
No Real Solution
16
Evaluate
A
C
.8
B
.08
D
No Real Solution
17
Evaluate
A
B
C
D
No Real Solution
Approximating
Square Roots
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Table of
Contents
Approximating a Square Root
Approximate
<
6
<
to the nearest integer
<
<
Identify perfect squares closest to 38
7
Take square root
Answer: Because 38 is closer to 36 than to 49,
nearest integer,
=6
is closer to 6 than to 7. So, to the
Approximate
to the nearest integer
<
<
<
<
Identify perfect squares closest to 70
Take square root
Identify nearest integer
18
Approximate
to the nearest integer
19
Approximate
to the nearest integer
20
Approximate
to the nearest integer
21
Approximate
to the nearest integer
22
Approximate
to the nearest integer
23
The expression
A
3 and 9
B
8 and 9
C
9 and 10
D
46 and 47
is a number between
Rational & Irrational
Numbers
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Table of
Contents
Rational & Irrational Numbers
is rational because the radicand (number under the radical) is a perfect
square
If a radicand is not a perfect square, the root is said to be irrational.
Ex:
Sort by the square root being rational or irrational.
24
A
Rational or Irrational?
Rational
B
Irrational
25
A
Rational or Irrational?
Rational
B
Irrational
26
A
Rational or Irrational?
Rational
B
Irrational
27
A
Rational or Irrational?
Rational
B
Irrational
28
A
Rational or Irrational?
Rational
B
Irrational
29
Which is a rational number?
A
B
C
D
p
30
Given the statement: “If x is a rational number,
x makes the statement false?
A
B
2
C
3
D
4
then
is irrational.”Which value of
Radical Expressions
Containing Variables
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Table of
Contents
Square Roots of Variables
To take the square root of a variable rewrite its exponent as the square of a
power.
=
=
Square Roots of Variables
If the square root of a variable raised to an even power has a variable raised to an odd
power for an answer, the answer must have absolute value signs. This ensures that the
answer will be positive.
By Definition...
Examples
Try These.
How many of these expressions will need an absolute value
simplified?
sign when
31
Simplify
A
B
C
D
32
Simplify
A
B
C
D
33
Simplify
A
B
C
D
34
Simplify
A
B
C
D
35
A
C
B
D
no real solution
Simplifying Non-Perfect Square
Radicands
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Table of
Contents
What happens when the radicand is not a perfect square?
Rewrite the radicand as a product of its largest perfect square factor.
Simplify the square root of the perfect square.
When simplified form still contains a radical, it is said to be irrational.
Try These.
Identifying the largest perfect square factor when simplifying radicals will result in the
least amount of work.
Ex:
Not simplified! Keep going!
Finding the largest perfect square factor results in less work:
Note that the answers are the same for both solution processes
36
Simplify
A
B
C
D
already in simplified form
37
Simplify
A
B
C
D
already in simplified form
38
Simplify
A
B
C
D
already in simplified form
39
Simplify
A
B
C
D
already in simplified form
40
Simplify
A
B
C
D
already in simplified form
41
Simplify
A
B
C
D
already in simplified form
42
Which of the following does not have an irrational simplified form?
A
B
C
D
2
43
Simplify
A
B
C
D
44
Simplify
A
B
C
D
45
Simplify
A
B
C
D
Simplify
46
A
B
C
D
Simplify
47
A
B
C
D
48
When
is written in simplest radical form, the result is
What is the value of k?
A
20
B
10
C
7
D
4
.
49
When
is expressed in simplest
form, what is the value of a?
A
6
B
2
C
3
D
8
Express −3 48 in simplest radical form.
Simplifying Roots of Variables
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Table of
Contents
Simplifying Roots of Variables
Remember, when working with square roots, an absolute
value sign is needed if:
the power of the given variable is even
and
the answer contains a variable raised to an odd power outside the radical
Examples of when absolute values are needed:
Simplifying Roots of Variables
Divide the exponent by 2. The number of times that 2 goes into the exponent becomes
the power on the outside of the radical and the remainder is the power of the radicand.
Note:
Absolute value signs are not needed because the radicand had an odd power to
start.
Example
Simplify
Only the y has an odd power on the outside of the
radical.
The x had an odd power under the radical so
no absolute value signs needed.
The m's starting power was odd, so it does
not require absolute value signs.
Simplify
A
B
C
D
Pull
50
51
Simplify
A
B
C
D
52
A
B
C
D
Simplify
53
A
B
C
D
Simplify
Operations with Radicals
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Table of
Contents
Adding or Subtracting Radicals
Radicals can be added and subtracted when they have like terms.
Like Terms means they have the same radicands.
Like Terms Unlike Terms
Identify the like terms
54
A
B
C
D
E
F
To add or subtract radicals, add or subtract the coefficients; the
radicand remains the same.
Examples.
Try These.
55
Simplify
A
B
C
D
Already Simplified
56
Simplify
A
B
C
D
Already Simplified
57
Simplify
A
B
C
D
Already Simplified
58
Simplify
A
B
C
D
Already Simplified
59
Simplify
A
B
C
D
Already Simplified
Radicals must be simplified before adding or subtracting
60
Simplify
A
B
C
D
Already in simplest form
61
Simplify
A
B
C
D
Already in simplest form
62
What is the sum of
A
B
7
C
9
D
29
and
?
63
What is the sum of
A
B
C
D
and
?
64
The expression
is equivalent to
A
B
C
D
10
-
65
Simplify
A
B
C
D
Already in simplest form
66
Which of the following expressions does not equal the other 3
expressions?
A
B
C
D
Multiplying Radicals
To multiply radicals, multiply the coefficients then multiply
possible.
the radicands. Simplify if
Multiplying Radicals
coefficient times coefficient and radicand times radicand
67
Multiply
A
B
C
D
Multiplying Radicals
After multiplying, check to see if radicand can be simplified.
68
Simplify
A
B
C
D
69
Simplify
A
B
C
D
70
A
B
C
D
Simplify
71
A
B
C
D
Simplify
Multiplying Polynomials Involving Radicals
1) Follow the rules for distribution.
2) Be sure to simplify radicals when possible and combine like terms.
Multiply and write in simplest form:
72
A
B
C
D
73
Multiply and write in simplest form:
A
B
C
D
74
Multiply and write in simplest form:
A
B
C
D
75
Multiply and write in simplest form:
A
B
C
D
76
Multiply and write in simplest form:
A
B
C
D
Rationalizing the Denominator
Which of these expressions has a rational denominator?
Rational
Denominator
Irrational
Denominator
A simplified fraction does not have a radical in the denominator.
The process of eliminating a radical in the denominator is called
"rationalizing the denominator".
To rationalize the denominator, you create an equivalent fraction by
multiplying the numerator & denominator by the denominator's
radical.
Examples.
Simplify
77
A
B
C
D
Already simplified
Simplify
78
A
B
C
D
Already simplified
Simplify
79
A
B
C
D
Already simplified
Simplify
80
A
B
C
D
Already simplified
Simplify
81
A
B
C
D
Already simplified
Pythagorean Theorem
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Table of Contents
Recall...
right triangle
is a triangle with a right angle.
hypotenuse
leg
leg
The sides form that right angle are the legs.
The side opposite the right angle is the hypotenuse.
The hypotenuse is also the longest side.
Pythagorean Theorem (R1)
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.
leg2 + leg2 = hypotenuse2
or
a2 + b2 = c2
c
a
b
Example
Find the length of the missing side of the right triangle.
x
9
12
Is the missing side a leg or the hypotenuse of the right triangle?
hypotenuse
x
9
12
92 + 122 = x2
81 + 144 = x2
225 = x2
15 = x
-15 is a extraneous solution, a distance can not equal a
negative number.
x = 15
Example
Find the length of the missing side.
x
20
28
Is the missing side a leg or the hypotenuse of the
right triangle?
leg
x
28
x2 + 202 = 282
x2 + 400 = 784
x2 = 384
x=8 6
20
82
The missing side is the ________ of the right triangle.
A
leg
B
hypotenuse
x
6
9
83
Find the length of the missing side.
x
6
9
84
The missing side is the _________ of the right triangle.
A
leg
x
B
hypotenuse
36
15
85
Find the length of the missing side.
x
36
15
The safe distance of the base of the ladder from a wall it leans against should be onefourth of the length of the ladder.
Thus, the bottom of a 28-foot ladder should
be 7 feet from the wall. How far up the wall
will a the ladder reach?
28 feet
?
7 feet
28 feet
?
7 feet
a2 + b2 = c2
72 + b2 = 282
49 + b2 = 384
b2 = 335
b 18.30
The ladder will reach 18.3 feet up the wall safely.
Try this...
The dimensions of a high school basketball court are 84' long and 50' wide. What
is the length of from one corner of the court to the opposite corner?
50
x
Answer
84
86
A NBA court is 50 feet wide and the length from one corner of the
court to the opposite corner is 106.5 feet. How long is the court?
(Round the answer to the nearest whole number)
A
94.03 feet
B
117.7 feet
C
118 feet
D
94 feet
Pythagorean Theorem Applications
The Pythagorean Theorem can also be used in figures that
contain right angles.
Example
Find the perimeter of the square.
18 cm
Psq = 4s
Before finding the perimeter of the square, we need to first find the
length of each side.
Remember, in a square all sides are congruent.
x
18 cm
x2 + x2 = 182
2x2 = 324
x2 = 162
x2 = 9 2
P = 4s
P = 4(9 2)
P = 36 2 cm
Example
Find the area of the triangle.
13 feet
13 feet
A =
1
bh
2
10 feet
The base of the triangle is given, but we need to find the height of
the triangle.
By definition, the altitude (or height) of an isosceles triangle is the
perpendicular bisector of the base.
13 feet
5 feet
5 feet
52 + h2 = 132
25 + h2 = 169
h2 = 144
h = 12
A = (10)(12)
A = (120)
A = 60 feet
13 feet
h
1
2
1
2
Try this...
Find the perimeter of the rectangle.
8 in
Prect = 2l + 2w
ANSWER
10 in
87
Find the area of the rectangle.
A
120 feet
B
84 feet
8 feet
C
46 inches
D
46 feet
88
Find the perimeter of the square. (Round to the nearest tenth)
A
25.46 cm
B
25.4 cm
C
25.5 cm
D
25.6 cm
89
Find the area of the triangle.
7 inches
7 inches
24 inches
Converse of the Pythagorean Theorem (R2)
If the square of the longest side of a triangle is equal to the sum of
the squares of the other two sides, then the triangle is a right
triangle.
B
If c2 = a2 + b2, then
ABC is a right triangle.
c
a
C
b
A
Example
Tell whether the triangle is a right triangle.
D
24
E
7
25
Remember c is the longest
side
c2 = a2 + b2
252 = 72 + 242
625 = 49 + 576
625 = 625
DEF is a right triangle.
F
Theorem (R3)
If the square of the longest side of a triangle is greater than the sum of
the squares of the other two sides, then the triangle is obtuse.
B
If c2 > a2 + b2, then
ABC is obtuse.
c
a
C
b
A
Theorem (R4)
If the square of the longest side of a triangle is less than the sum of the
squares of the other two sides, then the triangle is acute.
B
a
If c2 < a2 + b2, then
ABC is acute.
c
A
C
b
Example
Classify the triangle as acute, right, or obtuse.
15
13
17
c2 ? a2 + b2
172 ? 152 + 132
289 ? 225 + 169
289 < 394
The triangle is acute.
Example
Tell whether 12, 3, 3 15 represent the sides of a acute, right, or obtuse triangle.
First, we need to find the approximate value of 3 15, to determine if 3 15 or 12
is the longest side.
3 15 11.62, so 12 is the longest side.
122 ? 32 + (3 15)2
144 ? 9 + 135
144 = 144
The triangle is right.
90
Classify the triangle is acute, right, obtuse, or not a triangle.
A
acute
B
right
C
obtuse
D
not a triangle
12
15
11
91
A
Classify the triangle is acute, right, obtuse, or not a triangle.
acute
5
B
right
3
6
C
obtuse
D
not a triangle
92
Classify the triangle is acute, right, obtuse, or not a triangle.
A
acute
B
right
25
20
19
C
obtuse
D
not a triangle
93
Tell whether the lengths 35, 65, and 56 represent the sides of an
acute, right, or obtuse triangle.
A
acute
B
right
C
obtuse
94
Tell whether the lengths represent the sides of an acute, right, or
obtuse triangle.
A
acute triangle
B
right triangle
C
obtuse triangle
Review
If c2 = a2 + b2, then triangle is right.
If c2 > a2 + b2, then triangle is obtuse.
If c2 < a2 + b2, then triangle is acute.
Distance
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Contents
Computing the distance between two points in the plane is an application
of the Pythagorean Theorem for right triangles.
Computing distances between points in the plane is equivalent to finding
the length of the hypotenuse of a right triangle.
Relationship between the
Pythagorean Theorem & Distance Formula
The distance formula
calculates the distance
using point's coordinates.
The Pythagorean Theorem states a relationship
among the sides of a right triangle.
(x2, y2)
c2= a2 + b2
c
c
a
b
(x1, y1)
(x2, y1)
The Pythagorean Theorem is true for all right triangles. If we know the lengths of two
sides of a right triangle then we know the length of the third side.
Distance
The distance between two points, whether on a line or in a coordinate plane, is
computed using the distance formula.
The Distance Formula
The distance 'd' between any two points with coordinates
is given
(x2,
y2) by the formula:
(x1, y1)and
d=
Note: recall that all coordinates are (x-coordinate, y-coordinate).
Example
Calculate the distance
from Point K to Point I
(x1, y1)
(x2, y2)
Label the points - it does not matter
which one you label point 1 and point 2.
Your answer will be the same.
d=
Plug the coordinates into the distance formula
KI =
KI =
=
=
95
Calculate the distance from Point J to Point K
A
B
C
D
96
Calculate the distance from H to K
A
B
C
D
Calculate the distance from Point G to Point K
97
A
B
C
D
98
Calculate the distance from Point I to Point H
A
B
C
D
99
Calculate the distance from Point G to Point H
A
B
C
D
Trigonometric Ratios
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Table of Contents
Trigonometry - is a branch of mathematics
that deals with relationship of the sides and angles of triangles.
A trigonometric ratio is the ratio of the two lengths of a right triangle.
There are 3 ratios for each acute angle of a right triangle.
The ratios are called sine, cosine, and tangent abbreviated sin, cos, and tan respectively.
A
c
b
C
a
B
A
c
b
a
C
sinθ =
side opposite
hypotenuse
B
cosθ =
side adjacent
hypotenuse
CAH
SOH
tanθ =
side opposite
side adjacent
TOA
SOHCAHTOA
A
c
b
In each right triangle, there are 2 acute angles. In
the triangle to the left <A and <B are the acute
angles.
Let's look at <A.
Find the side opposite, side adjacent, hypotenuse.
C
a
B
A
The side opposite <A is a.
The hypotenuse is c.
The side adjacent (or next to) <A is b.
SOHCAHTOA
sinA =
side opposite <A
hypotenuse
cosA =
side adjacent <A
hypotenuse
tanA =
side opposite <A
side adjacent to <A
c
b
C
a
c
=
b
c
=
=
a
b
a
B
Example
D
10
6
Find the sin, cos, and tan of <F.
E
8
F
What is the side opposite, side adjacent, and the hypotenuse of the right
triangle?
D
10
6
E
F
8
DF is the hypotenuse. DE is the side opposite to < F.
EF is the side adjacent to <F.
sinF =
opp
hyp
6
= 10
=
tan F =
3
5
cosF =
opp
adj
=
6
8
=
3
4
adj
hyp
=
8
10
= 4
5
100
A
What is the side opposite to <J?
JL
J
B
LK
C
KJ
L
K
101
What is the hypotenuse of the triangle?
A
JL
B
LK
C
KJ
J
L
K
102
What is the side adjacent to <J?
A
JL
B
LK
C
KJ
J
L
K
103
A
What is the sinR?
9/13
Q
13
B
7/9
7
C
R
7/13
9
D
9/7
S
104
A
What is the cosR?
9/13
Q
13
B
7/9
7
C
D
R
7/13
9/7
9
S
105
What is the tanR?
A
9/13
B
7/9
C
Q
13
7
R
7/13
9
S
D
9/7
Using Trigonometric Ratios to find side length.
(You will need a calculator or trig table)
G
E
25o
x
12
M
When solving right triangles, you can use either acute angle to find the
answer.
G
E
25o
x
12
Referring to <G.
M
EM is the side opposite and GM is the hypotenuse.
The trig ratio that trig ratio uses the side opposite and
hypotenuse, is the sine function.
In the triangle, the length of GM is
given and EM is the side we need to
find.
G
E
25o
x
12
M
sin G =
sin25 =
(12)
.4226 =
x ≈ 5.07
EM
GM
x
12
x
12
(12)
G
E
25o
x
12
M
Referring to <M.
EM is the side adjacent and GM is the hypotenuse.
The trig ratio that uses the side adajacent and the hypotenuse, is the cosine function.
G
E
25o
x
12
M
(12)
cos M =
EM
GM
cos 65 =
x
12
.4226 =
x
12
x ≈ 5.07
(12)
C
70o
In the triangle, the length of CE is
given and EA is the side we need to
find.
10
20o
E
y
A
Referring to <C.
EA is the side opposite and CE is the side adjacent.
Referring to <A.
CE is the side opposite and EA is the side adjacent.
The trig ratio that uses the side opposite and the side adjacent, is the tangent function.
C
70o
10
20o
E
tanC =
tan70 =
(10)
2.747 =
EA
CE
y
A
tan A =
CE
EA
tan 20 =
10
y
(y) .3640 =
10
y
y
10
y
10
(10)
.3640y = 10
y ≈ 27.47
y ≈ 27.47
(y)
106
Evaluate sin60. Round to the nearest tenthousandth.
107
Evaluate cos45. Round to the nearest ten- thousandth.
108
Evaluate tan30. Round to the nearest ten- thousandth.
109
Using <B, which is the correct ratio needed to solve for x.
B
A
sin40 = 12/x
B
cos40 = x/12
C
12
x
tan40 = 12/x
50o
E
D
sin40 = x/12
D
110
Using <D, which is the correct ratio needed to solve for x.
B
A
sin50 = 12/x
B
cos50 = x/12
C
tan50 = 12/x
12
x
50o
E
D
sin50 = x/12
D
111
Using <J, which is the correct ratio needed to solve for y.
A
tan32 = x/11
B
cos32 = x/11
C
tan32 = 11/x
D
sin32 = 11/x
K
11
32o
J
x
L
112
Using <K, which is the correct ratio to solve for y.
A
tan58 = x/11
B
cos58 = x/11
K
11
C
tan58 = 11/x
32o
D
J
sin 58 = 11/x
x
L
113
Find the length of LM.
P
12
L
68o
M
114
Find the length of LP.
P
12
L
68o
M
Solving Right Triangles
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Table of Contents
To solve a right triangle means to find the length of each side and the measure of
each angle in the triangle.
When using trigonometric ratios to solve a right triangle, you need to know either
the length of 2 sides or the length of one side and the measure of one the acute
angles.
Remember
m<A + m<B + m<C = 180o
a2 + b2 = c2
SOHCAHTOA
In this section you will need to use the inverse trig function to solve the
equations. Just as the following are inverses and undo each other,
Addition Subtraction
Multiplication Division
Square Square Root
inverse
so does a trig ratio
and its inverse.
inverse
sinθ sin-1θ
inverse
cosθ cos-1θ
tanθ tan-1θ
inverse
inverse
inverse
115
Find sin-10.8. Round to the nearest hundredth.
116
Find tan-12.3. Round to the nearest hundredth.
117
Find cos-10.45. Round to the nearest hundredth.
9
A
B
15
C
In ABC we need to find the m<A, m<C and BC.
Referring to <C, AB is the side opposite and
AC is the hypotenuse
Referring to <A, AB is the side adjacent and
AC is the hypotenuse
Which functions should be used to find the m<C and m<A?
9
A
B
15
C
sinC =
AB
AC
sinC =
9
15
To find the m<A, use
the cos function.
cosA =
AB
AC
cosA =
9
15
sinC = 0.6
cos A = 0.6
sin-1C ≈ 36.87
cos-1A ≈53.13
m<C ≈36.87o
m<A ≈ 53.13o
CHECK
To find the m<C, use the sin
function.
9
A
B
15
C
Since two sides of the triangle is given, to find BC use the Pythagorean
Theorem.
a2 + b2 = c2
92 + x2 = 152
81 + x2 = 225
x2 = 144
x = 12
BC = 12
Try this...
Solve the right triangle. Round your answers to the
nearest hundredth.
R
24
7
Q
S
QS
= 25
Click
to
m<Q
=
73.74o
Reveal Answer
m<R = 16.26o
118
Find CE.
5
D
8
C
E
119
Find m<C.
5
D
8
C
E
120
Find the m<E.
5
D
8
C
E
Find the missing parts of the triangle.
A
15
64o
L
B
Referring to <L, AB is the side opposite and
AL is the hypotenuse.
Which trig function must be used?
A
15
64o
L
B
sin L =
AB
AL
sin64 =
z
15
.8988 =
z
15
z ≈ 13.48
AB ≈ 13.48
A
15
z
64o
L
m<L + m<A = 90o
64o + m<A = 90o
m<A = 26o
a2 + b2 = c2
a2 + (13.48)2 = 152
a2 + 181.79 = 225
a2 = 43.29
a ≈ 6.58
B
Try this...
Find the missing parts of the triangle.
R
37o
E
D
Click
to
RD
≈ 18.09
Reveal Answer
ED ≈ 14.36
m<R = 53o
121
Find the m<G.
L
A
18
G
122
Find AL.
L
A
18
G
123
Find the m<P.
P
A
49.19o
B
33.69o
C
41.81o
12
E
18
D
56.31o
N
124
Find RT.
A
10.44
B
12.45
S
T
8
40o
C
11.47
R
D
9.53