Question

In triangle ABC, if angle ACB is 90°, CD is perpendicular to AB, angle ACD is 30°, and AD is 6 cm, find the length of BC.

Answer 1

Using trigonometry and the Pythagorean theorem, the length of BC in the right-angled triangle ABC is determined to be √57 cm, approximately 7.55 cm.

Step-by-step explanation:

To find the length of BC in triangle ABC where angle ACB is 90° and CD is perpendicular to AB, with angle ACD being 30° and AD being 6 cm, we can use trigonometric ratios in right-angled triangles.

Step-by-Step Explanation:

1. Since angle ACD is 30° in the right-angled triangle ACD, and AD is the side adjacent to angle ACD, we can use the cosine function to find AC. The cosine of 30° is √3/2, therefore AC = (AD) / (cos 30°) = 6 / (√3/2) = 4√3 cm.
2. In triangle ACD, DC can be found using the sine function since DC is the side opposite angle ACD. The sine of 30° is 1/2, therefore DC = (AD) * (sin 30°) = 6 * (1/2) = 3 cm.
3. Since CD is perpendicular to AB and ACB is a right angle, triangle BCD is also a right-angled triangle with angle BCD being 90°. We can use the Pythagorean theorem to find BC. BC² = AC² + CD², hence BC = √(AC² + CD²) = √((4√3)^2 + 3²) = √(48 + 9) = √57 cm.

So, the length of BC is √57 cm, which is approximately 7.55 cm.