If none of the angles of a triangle is a right angle, the triangle is called oblique. All angles are...
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If none of the angles of a triangle is a right angle, the triangle is called oblique. All angles are acute Two acute angles, one obtuse angle
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Transcript of If none of the angles of a triangle is a right angle, the triangle is called oblique. All angles are...
If none of the angles of a triangle is a right angle, the triangle is called oblique.
All angles are acute
Two acute angles, one obtuse angle
To solve an oblique triangle means to find the lengths of its sides and the measurements of its angles.
FOUR CASES
CASE 1: One side and two angles are known (SAA or ASA).
CASE 2: Two sides and the angle opposite one of them are known (SSA).
CASE 3: Two sides and the included angle are known (SAS).
CASE 4: Three sides are known (SSS).
CASE 1: ASA or SAA
S
A
AASA
SA A
SAA
S
SA
CASE 2: SSA
S
SA
CASE 3: SAS
S
S
S
CASE 4: SSS
The Law of Sines is used to solve triangles in which Case 1 or 2 holds. That is, the Law of Sines is used to solve SAA, ASA or SSA triangles.
Law of Sines
For a triangle with sides and opposite
angles , , , respectively,
a b c, ,
sin sin sin a b c
Law of Sines
c
C
b
B
a
A sinsinsin
For a triangle with sides a, b, and c, and angles A, B, and C
Solve the triangle: = 30 (SAA) , , 70 5a
70
30
5
b c
180
30 70 180
80
sin sin305
70
b
b 5 70
309 40
sin
sin.
sin sin305
80
c
c 5 80
309 85
sin
sin.
Solve the triangle: = 20 (ASA) , , 60 12c
20
60
12
a
b
180
20 60 180 100
sin sin20 10012
a
a 12 20
1004 17
sin
sin.
sin sin60 10012
b
b 12 60
10010 55
sin
sin.
The area A of a triangle is
A bh12
where b is the base and h is the altitude drawn to that base.
h
b
a
sin ha
h a sin
A bh b a 12
12
sin 12
absin
The area A of a triangle equals one-half the product of two of its sides times the sine of its included angle.
A ab
A ac
A bc
121212
sin
sin
sin
Find the area of the triangle for which A a
c
5
7 70
,
, .
A ac12
sin
12
5 7 70sin
16 44.
Find the area of a triangle ABC if a = 5, C = 65 degrees,
and B = 45 degrees.
Solve the triangle: (SSA)c b 5 3 50, ,
sin sin503 5
sinsin 5 50
3
sin . 128
No triangle with the given measurements!
3
5
50a
Solve the triangle: (SSA)b c 5 3 30, ,
30
5
3
a
sin sin305 3
sinsin
. 3 305
0 3
1 17 5 . 2 162 5 .
2 30 162 5 192 5 180 . .
180 30 17 5 132 5 . .
sin . sin132 5 305
a
a 5 132 5
307 37
sin .
sin.
a b c
7 37 5 3
132 5 30 17 5
. , ,
. , , .
Solve the triangle: (SSA)b c 8 10 45, ,
sin sin458 10
sinsin
. 10 458
0 88
1 262 1 117 9 . . or
45 62 1 180 . 45 117 9 180 .
Two triangles!!
Triangle 1: 1 62 1 .
1 180 45 62 1 72 9 . .
sin . sin72 9 4581
a
a18 72 9
4510 81 sin .
sin.
a b c1
1 1
10 81 8 10
72 9 45 62 1
. , ,
. , , .
Triangle 2: 2 117 9 .
2 180 45 117 9 17 1 . .
sin . sin17 1 4582
a
a28 17 1
453 33 sin .
sin.
a b c2
2 2
3 33 8 10
17 1 45 117 9
. , ,
. , , .
Lesson Overview 5-6B
5-Minute Check Lesson 5-7A
Lesson Overview 5-7A
Lesson Overview 5-7B
5-Minute Check Lesson 5-8A
Heron’s Formula
The area A of a triangle with sides a, b, and c is
A s s a s b s c
where s a b c 12
.
Find the area of a triangle whose sides are
5, 8, and 11.
Let a b c 5 8 11, ,
s a b c 12
12
5 8 11 12
A s s a s b s c
12 12 5 12 8 12 11
336 4 21
Lesson Overview 5-6A
