Question

In trapezoid ABCD where AD¯ ∥ BC¯, BC = 5 cm, m∠ACD = m∠ABC = 90°, m∠BAC = 30°, what is the length of AD¯?

(A) 5 cm
(B) 5√3 cm
(C) 10 cm
(D) 10√3 cm

Answer 1

The length of AD¯ is 5√10 cm.

Step-by-step explanation:

In trapezoid ABCD, AD¯ ∥ BC¯, BC = 5 cm, m∠ACD = m∠ABC = 90°, m∠BAC = 30°. To find the length of AD¯, we can use the cosine rule.

In triangle ABC, using the cosine rule: AB² = AC² + BC² - 2(AC)(BC)cos(90°)

Since AD¯ and BC¯ are parallel, triangle ABC and triangle ACD are similar. Therefore, AB = AD.

Substituting the values, we have:
AD² = AC² + BC² - 2(AC)(BC)cos(90°)

AD² = AC² + BC²
AD² = (3R)² + 5²
AD² = 9R² + 25

The length of AD¯ is √(9R² + 25).

Given that BC = 5 cm, the length of AD¯ is therefore √(9*(5²) + 25) = √(225 + 25) = √250 = 5√10 cm.