Question

A triangle is shown with two sides labeled x centimeters and 14 centimeters. The angle created by those sides is labeled 62 degrees. The angle across from the side measuring x centimeters is labeled 41 degrees.
Question
Use the image to answer the question.
What is the length of the side labeled x cm?
(1 point)
Responses

18.8 cm
18.8 cm

9.4 cm
9.4 cm

10.4 cm
10.4 cm

12.7 cm
12.7 cm
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Answer 1

To find the length of side \( x \) in the triangle, we can use the Law of Sines. The Law of Sines states that the ratio of the lengths of the sides of a triangle to the sine of the opposite angles is constant:

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

In this triangle, we have:

- Side \( a = x \) (opposite angle \( A \))
- Side \( b = 14 \) cm (opposite angle \( B \))
- Angle \( A = 41^\circ \)
- Angle \( B = 62^\circ \)

First, let's calculate angle \( C \):

\[
C = 180^\circ - A - B = 180^\circ - 41^\circ - 62^\circ = 77^\circ
\]

Next, we can set up the ratio using the Law of Sines:

\[
\frac{x}{\sin(41^\circ)} = \frac{14}{\sin(62^\circ)}
\]

Now, we can solve for \( x \):

\[
x = \frac{14 \cdot \sin(41^\circ)}{\sin(62^\circ)}
\]

Before substituting values, we will calculate \( \sin(41^\circ) \) and \( \sin(62^\circ) \):

- \( \sin(41^\circ) \approx 0.6561 \)
- \( \sin(62^\circ) \approx 0.8746 \)

Substituting these values in gives:

\[
x = \frac{14 \cdot 0.6561}{0.8746} \approx \frac{9.2254}{0.8746} \approx 10.54 \text{ cm}
\]

So, rounding to one decimal place, the length of the side labeled \( x \) cm is approximately **10.5 cm**. However, it seems the answer options do not include this value.

The closest answer from the options given is **10.4 cm**. Therefore, the length of the side labeled \( x \) cm is:

**10.4 cm**.