Question

From the adjoining figure, find the length of CD.

Answer 1

Given: In triangle ACD, ∠C = 60°, ∠D = 30°, so ∠A = 90°. AB is the perpendicular from A to CD with AB = 45 cm.

Step-wise calculation:

1. In right ΔACD right at A the angles are 30° – 60° – 90°. 

Let AC = k (side opposite 30°), AD = `ksqrt(3)` (side opposite 60°) and CD = 2k (hypotenuse).

2. The altitude from the right angle to the hypotenuse satisfies

`AB = (AC xx AD)/(CD)` 

`AB = (k xx ksqrt(3))/(2k)` 

= `(ksqrt(3))/2`

3. Solve for k using AB = 45 cm:

`(ksqrt(3))/2 = 45`

⇒ `k = (2 xx 45)/sqrt(3)` 

= `90/sqrt(3)`

4. CD = 2k

= `2 xx (90/sqrt(3))` 

= `180/sqrt(3)` 

= `(180sqrt(3))/3`

= `60sqrt(3)` cm

CD = `60sqrt(3)` cm ≈ 103.92 cm.