Question

If an oblique triangle has side b=12 mm long, side c=14 mm long, ∠A=35 degrees, then what is the measure of side a to the nearest millimeter?

Answer 1

To find the length of side \( a \) in an oblique triangle (non-right triangle), we can use the Law of Cosines, which states:

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

Given:
- \( b = 12 \) mm
- \( c = 14 \) mm
- \( A = 35^\circ \)

First, we need to calculate \( \cos(35^\circ) \). Using a calculator, we find:

\[
\cos(35^\circ) \approx 0.8192
\]

Now we can substitute the values into the Law of Cosines formula:

\[
a^2 = 12^2 + 14^2 - 2 \cdot 12 \cdot 14 \cdot \cos(35^\circ)
\]

Calculating \( 12^2 \) and \( 14^2 \):

\[
12^2 = 144
\]
\[
14^2 = 196
\]

Now substituting:

\[
a^2 = 144 + 196 - 2 \cdot 12 \cdot 14 \cdot 0.8192
\]

Now calculating the product:

\[
2 \cdot 12 \cdot 14 = 336
\]
\[
336 \cdot 0.8192 \approx 275.2
\]

Now substituting back into the equation for \( a^2 \):

\[
a^2 = 144 + 196 - 275.2
\]
\[
a^2 = 340 - 275.2
\]
\[
a^2 \approx 64.8
\]

Finally, taking the square root to find \( a \):

\[
a \approx \sqrt{64.8} \approx 8.06
\]

Rounding to the nearest millimeter:

\[
a \approx 8 \text{ mm}
\]

Thus, the measure of side \( a \) is approximately **8 mm**.