Question

A triangle with three sides labeled as 1.6 yards, x yards, and 1.1 yards. The angle opposite the side labeled x yards is labeled 78 degrees.
Question
Use the image to answer the question.
Solve for x in the triangle.
(1 point)
Responses

1.7
1.7

2.6
2.6

2.7
2.7

3.0

Answer 1

To determine the value of \( x \) in the triangle with sides 1.6 yards, \( x \) yards, and 1.1 yards, and an angle of 78 degrees opposite to the side \( x \), we can use the Law of Cosines. The Law of Cosines states:

\[
c^2 = a^2 + b^2 - 2ab \cos(C)
\]

In this case, we can assign:

- \( a = 1.6 \) yards
- \( b = 1.1 \) yards
- \( C = 78^\circ \) (angle opposite to side \( x \))
- \( c = x \)

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

\[
x^2 = (1.6)^2 + (1.1)^2 - 2(1.6)(1.1) \cos(78^\circ)
\]

Calculating each term step by step:

1. Calculate \( (1.6)^2 \):
\[
(1.6)^2 = 2.56
\]

2. Calculate \( (1.1)^2 \):
\[
(1.1)^2 = 1.21
\]

3. Calculate \( 2(1.6)(1.1) \):
\[
2(1.6)(1.1) = 3.52
\]

4. Calculate \( \cos(78^\circ) \) (you can use a calculator for this):
\[
\cos(78^\circ) \approx 0.2079
\]

5. Calculate \( 3.52 \times \cos(78^\circ) \):
\[
3.52 \times 0.2079 \approx 0.7311
\]

Putting it all together into the equation:

\[
x^2 = 2.56 + 1.21 - 0.7311
\]

6. Combine the values:
\[
2.56 + 1.21 - 0.7311 \approx 3.0389
\]

7. So, we have:
\[
x^2 \approx 3.0389
\]

Finally, take the square root of both sides:
\[
x \approx \sqrt{3.0389} \approx 1.74
\]

Since the options given are 1.7, 2.6, 2.7, and 3.0, the closest answer to 1.74 is:

**1.7**