AnswersFall 24 GMS Geometry Concepts and Connections BLaw of Sines (Sec 8-3)

Law of Cosines Answers

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Consider the proof.Given: In △ABC, BD ⊥ ACProve: the formula for the law of cosines, a2 = b2 + c2 – 2bccos(A) Statement Reason1.In △ABC, BD ⊥ AC1.given2.In △ADB, c2 = x2 + h22.Pythagorean thm. 3.In △BDC, a2 = (b – x)2 + h23.Pythagorean thm. 4.a2 = b2 – 2bx + x2 + h24.prop. of multiplication5.a2 = b2 – 2bx + c25.substitution6.In △ADB, cos(A) = 6.def. cosine7.ccos(A) = x7.mult. prop. of equality8.a2 = b2 – 2bccos(A) + c28.?9.a2 = b2 + c2 – 2bccos(A)9.commutative propertyWhat is the missing reason in Step 8?

Question illustration
A
Pythagorean theorem
B
definition of cosine
C
substitution
D
properties of multiplication
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On which triangle can the law of cosines be applied once to find an unknown angle measure?Law of cosines: a2 = b2 + c2 – 2bccos(A)

A
Triangle X W Y is shown. Angle X Y W is a right angle. The length of hypotenuse W X is y, the length of X Y is a, and the length of W Y is 9.
Option A
B
Triangle W X Y is shown. Sides W X and X Y are congruent. The length of W Y is 7.
Option B
C
Triangle W X Y is shown. Angle X Y W is 76 degrees. The length of X Y is 5, the length of W Y is 10, and the length of W X is y.
Option C
D
Triangle W X Y is shown. The length of W X is 10, the length of X Y is 8, and the length of W Y is 16.
Option D
3

Law of cosines: a2 = b2 + c2 – 2bccos(A)

Question illustration
A
70°
B
77°
C
80°
D
85°
4

The law of cosines is used to find the measure of .

Question illustration
A
41º
B
47º
C
51º
D
57º
5

Law of cosines: a2 = b2 + c2 – 2bccos(A)

Question illustration
A
58°
B
64°
C
68°
D
73°
6

Law of cosines: a2 = b2 + c2 – 2bccos(A)

Question illustration
A
62 = p2 + 82 – 2(p)(8)cos(39°)
B
p2 = 62 + 82 – 2(6)(8)cos(39°)
C
82 = 62 + p2 – 2(6)(p)cos(39°)
D
p2 = 62 + 62 – 2(6)(6)cos(39°)
7

The law of cosines is used to find the measure of .

Question illustration
A
44º
B
49º
C
54º
D
59º
8

Law of cosines: a2 = b2 + c2 – 2bccos(A)

Question illustration
A
19°
B
26°
C
30°
D
33°
9

On which triangle can the law of cosines be used to find the length of an unknown side?Law of cosines: a2 = b2 + c2 – 2bccos(A)

A
Triangle Q R S is shown. The length of Q R is s, the length of R S is q, and the length of Q S is 12. Angle R Q S is 36 degrees, angle Q S R is 57 degrees, and angle S R Q is 87 degrees.
Option A
B
Triangle Q R S is shown. The length of Q R is s, the length of R S is 7, and the length of Q S is 12. Angle R S Q is 57 degrees.
Option B
C
Triangle Q R S is shown. The length of Q R is s, the length of R S is 7, and the length of Q S is b. Angle R S Q is 57 degrees and angle S Q R is 36 degrees.
Option C
D
Triangle Q R S is shown. The length of Q R is s, the length of R S is q, and the length of Q S is 12. Angle R Q S is 36 degrees and angle Q S R is 57 degrees.
Option D
10

Abby used the law of cosines for KMN to solve for k.k2 = 312 + 532 – 2(31)(53)cos(37°)Law of cosines: a2 = b2 + c2 – 2bccos(A)

Question illustration
A
mK = 37° and n = 31
Option A
B
mK = 37° and k = 31
Option B
C
mN = 37° and n = 31
Option C
D
mN = 37° and k = 31
Option D

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