Question

If oblique ΔABC
has side b=32
in., side c=45
in., and included ∠A=56°
, then what is the length of side a? (1 point)
Responses

49 in.
49 in.

1,439 in.
1,439 in.

38 in.
38 in.

2,440 in.

Answer 1

To find the length of side \( a \) in triangle \( ABC \), we can use the Law of Cosines, which states:

\[
a^2 = b^2 + c^2 - 2bc \cdot \cos(A)
\]

Where:
- \( a \) is the side opposite angle \( A \),
- \( b = 32 \) in,
- \( c = 45 \) in,
- \( A = 56^\circ \).

Now we can plug in the values:

1. Calculate \( b^2 \) and \( c^2 \):
\[
b^2 = 32^2 = 1024
\]
\[
c^2 = 45^2 = 2025
\]

2. Calculate \( 2bc \):
\[
2bc = 2 \cdot 32 \cdot 45 = 2880
\]

3. Calculate \( \cos(56^\circ) \):
\[
\cos(56^\circ) \approx 0.5592
\]

4. Now substitute everything back into the Law of Cosines formula:
\[
a^2 = 1024 + 2025 - 2880 \cdot 0.5592
\]
Calculate \( 2880 \cdot 0.5592 \):
\[
2880 \cdot 0.5592 \approx 1619.78
\]

5. Now substitute this value:
\[
a^2 = 1024 + 2025 - 1619.78 \approx 1024 + 2025 - 1619.78 = 1429.22
\]

6. Taking the square root to find \( a \):
\[
a \approx \sqrt{1429.22} \approx 37.8
\]

So, rounding this gives approximately \( 38 \) in.

**The correct response is:** **38 in.**