To find the length of side $e$ in $\triangle DEF$, we can use the Law of Sines. The Law of Sines states:

$\frac{e}{\sin(\angle D)} = \frac{f}{\sin(\angle E)}$

Given:

• $f = 960$ inches
• $\angle D = 101^\circ$
• $\angle E = 27^\circ$

Step 1: Find the missing angle $\angle F$

Since the sum of the angles in a triangle is $180^\circ$:

$\angle F = 180^\circ - \angle D - \angle E = 180^\circ - 101^\circ - 27^\circ = 52^\circ$

Step 2: Apply the Law of Sines

Now, we can use the Law of Sines to find $e$:

$\frac{e}{\sin(27^\circ)} = \frac{960}{\sin(52^\circ)}$

Step 3: Solve for $e$

First, calculate the sine values:

$\sin(27^\circ) \approx 0.4540$ $\sin(52^\circ) \approx 0.7880$

Now, substitute these values into the equation:

$\frac{e}{0.4540} = \frac{960}{0.7880}$

$e = \frac{960 \times 0.4540}{0.7880} \approx \frac{435.84}{0.7880} \approx 552.9 \text{ inches}$

So, the length of side $e$ is approximately 552.9 inches.

Would you like to ask anything else or need more details?

Here are 5 related questions:

1. How can we use the Law of Cosines to find side lengths?
2. How do you find the area of a triangle given two sides and an included angle?
3. What happens when an angle in a triangle exceeds 90 degrees?
4. How does the Law of Sines differ when solving for angles instead of sides?
5. What is the importance of finding the smallest angle first in trigonometric calculations?

Tip: Always double-check angle measures to ensure they add up to 180° when using the Law of Sines or Cosines.