In triangle ABC, if AB = 10 cm, AC = 6 cm, and angle A = 30°, what is the length

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Question: In triangle ABC, if AB = 10 cm, AC = 6 cm, and angle A = 30°, what is the length of side BC?

Options:

Correct Answer: 7 cm

Solution:

Using the Law of Cosines: BC² = AB² + AC² - 2 * AB * AC * cos(A) = 10² + 6² - 2 * 10 * 6 * (√3/2) = 100 + 36 - 60√3. BC = √(100 + 36 - 60√3) ≈ 7 cm.

In triangle ABC, if AB = 10 cm, AC = 6 cm, and angle A = 30°, what is the length of side BC?

In triangle ABC, if AB = 10 cm, AC = 6 cm, and angle A = 30°, what is the length of side BC?

Step 1: Identify the sides and angle of triangle ABC. We have AB = 10 cm, AC = 6 cm, and angle A = 30°.
Step 2: Recall the Law of Cosines formula: BC² = AB² + AC² - 2 * AB * AC * cos(A).
Step 3: Substitute the known values into the formula: BC² = 10² + 6² - 2 * 10 * 6 * cos(30°).
Step 4: Calculate the squares: 10² = 100 and 6² = 36.
Step 5: Find cos(30°). The value of cos(30°) is √3/2.
Step 6: Substitute the values into the equation: BC² = 100 + 36 - 2 * 10 * 6 * (√3/2).
Step 7: Simplify the equation: BC² = 100 + 36 - 60√3.
Step 8: Combine the numbers: BC² = 136 - 60√3.
Step 9: To find BC, take the square root: BC = √(136 - 60√3).
Step 10: Use a calculator to approximate the value: BC ≈ 7 cm.

Law of Cosines – This theorem relates the lengths of the sides of a triangle to the cosine of one of its angles, allowing for the calculation of an unknown side when two sides and the included angle are known.
Trigonometric Functions – Understanding how to use trigonometric functions, specifically cosine, to solve for unknown lengths in triangles.