Download Right Triangle Notes Packet

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Geometry/Trigonometry
Unit 7: Right Triangle Notes
(1)
Page 430 #1 – 15
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Page 430 – 431 #16 – 23, 25 – 27, 29 and 31
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Page 437 – 438 #1 – 8, 9 – 19 odd
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Page 437 – 439 #10 – 20 Even, 23, and 29
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Page 444 #1 – 14
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Page 444 – 445 #15 – 26, 30 – 33
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Page 451 #1 – 8
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Page 451 – 452 #9 – 20
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Page 457 #1 – 10
Name:
Date:
Period:
(10) Page 457 – 458 #11 – 28
(11)
Page 465 #1 – 18
(12) Page 465 – 466 #19 – 29 Odd, 34 – 41 all
(13) Page 469 – 471 #1 – 8, 12, 13, 15, 16, 18 – 24 Even, 29 – 32, 39 – 44
(14) Page 473 #1 – 21
Geometry Notes 9.1 Exploring Right Triangles
All right triangles are either __________________________
To prove that two right triangles are similar you can show that _____________________________ of
one of the triangles is _________________________ to______________ of the ___________________
of the other triangle.
Theorem 9.1 (Use 3x5 Card to Explore): If the altitude is drawn to the hypotenuse of a right triangle,
then the two triangles formed are similar to the original triangle and to each other.
Therefore:
. What triangles are similar?
Theorem 9.2: In a right triangle, the length of the altitude from the right angle to the hypotenuse is the
geometric mean of the lengths of the two segments of the hypotenuse.
Theorem 9.3: In a right triangle, the altitude from the right angle to the hypotenuse divides the
hypotenuse into two segments. Each leg of the right triangle is the geometric mean of the hypotenuse
and the segment of the hypotenuse that is adjacent to the leg.
and
Geometry Notes 9.2 The Pythagorean Theorem
Theorem 9.4 The Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse is
equal to the sum of the squares of the lengths of the legs.
A set of three positive integers, a, b and c, that satisfy the equation
is a Pythagorean Triple.
(ex1)
3, 4, 5
(ex2)
_____________________
(ex3)
__________________
(ex4)
5, 12, 13
(ex5)
______________________
(ex6)
__________________
(ex7)
7, 24, 25
(ex8)
______________________
(ex9)
__________________
Geometry Notes 9.3 The Converse of the Pythagorean Theorem
Theorem 9.5 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 two shorter sides, then the triangle is a right triangle.
Theorem 9.6: 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 two shorter sides, then the triangle is acute.
If a2 + b2 > c2 or c2 < a2 + b2, then the triangle is an acute triangle
Theorem 9.7: 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 two shorter sides, then the triangle is obtuse.
If a2 + b2 < c2 or c2 > a2 + b2, then the triangle is an obtuse triangle
Geometry Notes 9.4 Special Right Triangles
45-45-90
Theorem 9.8 – In a 45-45-90 triangle, the hypotenuse is √ times as long as each leg. That is, the sidelength ratios of leg:leg:hyp are 1:1:√ .
E1. Find the missing values
E2. Find the missing values
P1. Find the missing values
P2. Find the missing values
30-60-90
Theorem 9.9 – In a 30-60-90 triangle, the hypotenuse is twice as long as the shorter leg and the longer
leg is √ as long as the shorter leg. That is, the side-length ratios of short leg : long leg : hyp is 1:√ :2
E1. Find t he missing values
E2. Find the missing values
E3. Find the missing values
P1. Find the missing values
P2. Find the missing values
P3. Find the missing values
Geometry Notes 9.5 Trigonometric Ratios
_______________________________ – measurement of triangles
From <A:
From <B:
B
B
_______________
c_______
_______________
a
_______
_________________
c
____
_______________ a
_______
C
A
C
b
_____________________
A
_______________
b
_______
Trigonometric Ratio – a ratio of the lengths of two sides of a triangle.
Sine (sin), Cosine (cos) and Tangent (tan) are the_______ basic ______________________ ratios
Ex1.
Find each trigonometric ratio:
(a)
sin A= _______
(b)
cos A=_______
(c)
tan A=_______
(d)
sin B=_______
(e)
cos B=_______
(f)
tan B=_______
B
5
C
B
5
13
A
12
13
C
12
Find the missing side or angle and round to the nearest tenth
E2.
E3.
E4.
E5.
A
Geometry Notes 9.6 Extended Application: Solving Right Triangles:
Solving a right triangle means to find the measure of __________________________ of the triangle and
__________________________ of the triangle. In other words all __________parts.
You can solve a right triangle if you know:
A.
B.
(1)
Two __________ lengths OR
(2)
One ____________________ length and one ______________________ measure
How to find an angle measure of a right triangle given two side measures
E1.
Sin A = .2548
E2.
Cos A = .5624
E3.
Tan A =
P1.
Sin A = .6895
P2.
Cos A = .4156
P3.
Tan A =
How to solve a right triangle given two side lengths
E4.
B
P4.
13
C
C.
B
5
2
A
12
C
A
How to solve a right triangle given one side length and an acute angle measure
E5.
P5.
B
B
3
C
°
7
A
C
54°
A
Geometry Notes Law of Sines and Law of Cosines:
E1. Solve the triangle (AAS)
E2. Solve the triangle (ASA)
E3. Solve the triangle (SSA – One Triangle) – The ambiguous case
E4. Solve the triangle (SSA – No Triangle) – The ambiguous case
E5. Solve the triangle (SSA – Two Triangles) – The ambiguous case
E6. Solve the triangle (SAS)
E7. Solve the triangle (SSS)