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

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