Find the length of side b.
Given:
- Angle A = 23°
- Angle C = 125°
- Side length between B and C = 7
Find the length of side b.
Round to the nearest tenth.
Answer (2)
To find the length of side b in the given triangle, we will use the Law of Sines. The Law of Sines states that for any triangle A BC , the ratio of the length of a side to the sine of its opposite angle is the same for all three sides. Mathematically, it is represented as:
sin A a = sin B b = sin C c
Given:
Angle A = 2 3 ∘
Angle C = 12 5 ∘
Side c = 7 (the side between B and C)
First, we need to find Angle B :
B = 18 0 ∘ − A − C = 18 0 ∘ − 2 3 ∘ − 12 5 ∘ = 3 2 ∘
Now, using the Law of Sines, we can find side b :
sin B b = sin C c
⇒ b = sin C c ⋅ sin B
Substituting the known values:
b = sin 12 5 ∘ 7 ⋅ sin 3 2 ∘
Using a calculator to find the sine values:
sin 3 2 ∘ ≈ 0.5299
sin 12 5 ∘ ≈ 0.8192
Plugging these values in, we get:
b ≈ 0.8192 7 ⋅ 0.5299
b ≈ 0.8192 3.7093 ≈ 4.5
Therefore, the length of side b is approximately 4.5 when rounded to the nearest tenth.
To find the length of side b, we used the Law of Sines after calculating angle B to be 32°. By substituting the known values into the Law of Sines, we calculated the length of side b to be approximately 4.5. Therefore, b ≈ 4.5 .
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