Question

In ΔABC , find the length, to the nearest tenth of an inch, of side a if b=7 in., c=10 in., and the angle formed by these sides measures 72 degrees

Answer 1

To find the length of side \( a \) in triangle \( \Delta ABC \) where \( b = 7 \) inches, \( c = 10 \) inches, and angle \( A \) (the angle opposite side \( a \)) measures \( 72^\circ \), we can apply the Law of Cosines.

The Law of Cosines states that:

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

Substituting the known values:

- \( b = 7 \)
- \( c = 10 \)
- \( A = 72^\circ \)

we can calculate \( a \):

\[
a^2 = 7^2 + 10^2 - 2 \cdot 7 \cdot 10 \cdot \cos(72^\circ)
\]

Calculating \( 7^2 \) and \( 10^2 \):

\[
7^2 = 49
\]
\[
10^2 = 100
\]

Now, substituting these values into the equation:

\[
a^2 = 49 + 100 - 2 \cdot 7 \cdot 10 \cdot \cos(72^\circ)
\]
\[
a^2 = 149 - 140 \cdot \cos(72^\circ)
\]

Next, we need to find \( \cos(72^\circ) \). The cosine of \( 72^\circ \) is approximately \( 0.309 \) (you can use a calculator or cosine table for precision):

\[
a^2 = 149 - 140 \cdot 0.309
\]
\[
a^2 = 149 - 43.26
\]
\[
a^2 \approx 105.74
\]

Now, we calculate the square root to find \( a \):

\[
a \approx \sqrt{105.74} \approx 10.3
\]

Thus, the length of side \( a \) to the nearest tenth of an inch is:

\[
\boxed{10.3 \text{ in}}
\]