High School

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Solve each triangle ABC given that:

- Angle A = 37.9 degrees
- Side a = 3.7
- Side c = 9.7

Find the missing values. Which of the following options is correct?

A. There is only one possible solution for the triangle.

Answer :

Final answer:

Given angle A, side a, and side c, Law of Cosines finds missing side b, then Law of Sines with two sides and one angle uniquely determines the triangle (Option A).

Option a.

Explanation:

There is only one possible solution for the triangle ABC based on the given information.

Here's why:

Law of Cosines: We can use the Law of Cosines to solve for the missing side (b) and subsequently the missing angle (B). The Law of Cosines states:

b² = a² + c² - 2ac * cos(A)

Known Values:

A = 37.9 degrees (angle)

a = 3.7 (side)

c = 9.7 (side)

Solve for b:

Plugging in the values:

b² = 3.7² + 9.7² - 2 * 3.7 * 9.7 * cos(37.9°)

Solving for b using a calculator, we get:

b ≈ 8.46

Law of Sines: Once we have two sides (a and b) and an angle (A), we can use the Law of Sines to solve for the remaining angle (B). The Law of Sines states:

a / sin(A) = b / sin(B) = c / sin(C)

Solve for B:

Using the known values and the calculated value of b:

3.7 / sin(37.9°) = 8.46 / sin(B)

Solving for B using a calculator, we get:

B ≈ 64.7 degrees

Since we have all the angles (A, B, and C = 180 - A - B) and two sides (a and b), the triangle is uniquely determined.

Therefore, the correct answer is:

A. There is only one possible solution for the triangle.

Option a.

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