Question

Question
Use the image to answer the question.

Right triangle upper A upper B upper C has a small square box at angle B. The side opposite of each angle is labeled as the lowercase letter of the angle name.

Use the Law of Sines to solve the following problem. If ∠C=35 degrees
and side c=42 yd.
, how long is side a to the nearest yard?

(1 point)
Responses

64 yd.
64 yd.

60 yd.
60 yd.

29 yd.
29 yd.

43 yd.

Answer 1

To use the Law of Sines to solve for side \(a\) in triangle \(ABC\), we can start with the information given:

- \( \angle C = 35^\circ \)
- \( c = 42 \) yd

From the Law of Sines, we know:

\[
\frac{a}{\sin A} = \frac{c}{\sin C}
\]

Rearranging this gives:

\[
a = c \cdot \frac{\sin A}{\sin C}
\]

To proceed, we need to find angle \(A\) and then apply the Law of Sines accordingly.

### Step 1: Calculate \(B\)

Since it's a right triangle, we know:

\[
A + B + C = 180^\circ
\]

Given that \( \angle B = 90^\circ \):

\[
A + 90^\circ + 35^\circ = 180^\circ
\]

So:

\[
A = 180^\circ - 90^\circ - 35^\circ = 55^\circ
\]

### Step 2: Calculate \(a\)

Now we have:

- \(A = 55^\circ\)
- \(C = 35^\circ\)
- \(c = 42\) yd

Plugging these values into the Law of Sines equation, we have:

\[
a = 42 \cdot \frac{\sin(55^\circ)}{\sin(35^\circ)}
\]

### Step 3: Calculate the sines

Using a calculator (make sure it’s in degrees):

- \( \sin(55^\circ) \approx 0.8192\)
- \( \sin(35^\circ) \approx 0.5736\)

### Step 4: Plug in the values

Now substituting these in:

\[
a = 42 \cdot \frac{0.8192}{0.5736} \approx 42 \cdot 1.4281 \approx 60.0
\]

### Step 5: Round and finalize

Therefore, rounding to the nearest yard, we obtain:

\[
a \approx 60 \text{ yd}
\]

Thus, the length of side \(a\) is:

**60 yd.**