Question

In triangle ABC, angle A is 30 degrees, angle B is 45 degrees, and side a (opposite angle A) is 10 units long. What is the length of side b (opposite angle B)?(1 point)
Responses

10.0
10.0

14.14
14.14

15.0
15.0

11.18

Answer 1

To find the length of side \( b \) (opposite angle \( B \)) in triangle \( ABC \), we can apply the Law of Sines. The Law of Sines states that:

\[
\frac{a}{\sin(A)} = \frac{b}{\sin(B)}
\]

Given:
- \( a = 10 \) (length of side opposite to angle A),
- \( A = 30^\circ \),
- \( B = 45^\circ \).

We first need to find \( \sin(A) \) and \( \sin(B) \):
- \( \sin(30^\circ) = \frac{1}{2} \),
- \( \sin(45^\circ) = \frac{\sqrt{2}}{2} \).

Now, we can set up the equation using the Law of Sines:

\[
\frac{10}{\sin(30^\circ)} = \frac{b}{\sin(45^\circ)}
\]

Substituting the sine values:

\[
\frac{10}{\frac{1}{2}} = \frac{b}{\frac{\sqrt{2}}{2}}
\]

This simplifies to:

\[
20 = \frac{b}{\frac{\sqrt{2}}{2}}
\]

To solve for \( b \), multiply both sides by \( \frac{\sqrt{2}}{2} \):

\[
b = 20 \cdot \frac{\sqrt{2}}{2} = 10\sqrt{2}
\]

Calculating \( 10\sqrt{2} \):

\[
b \approx 10 \cdot 1.414 \approx 14.14
\]

Therefore, the length of side \( b \) is approximately \( 14.14 \).

**The answer is:** \( 14.14 \)