Question

From the adjoining figure, find the length of AC.

Answer 1

Given:

AE = 18 cm (AE ⟂ AC) and CD = 18 cm (CD ⟂ AC).

At B the left slant BE makes 60° with the base and the right slant BD makes 30° with the base.

Hence, triangles ABE and BCD are right triangles with acute angles 30° and 60° (30-60-90 triangles).

Step-wise calculation:

1. In right ΔABE (∠A = 90°, ∠B = 60°):

AE is opposite 60°.

For a 30-60-90 triangle, sides opposite 30°, 60°, 90° are k, `ksqrt(3)`, 2k respectively.

`AE = ksqrt(3)` 

= 18 

⇒ `k = 18/sqrt(3)` 

= `6sqrt(3)`

So, AB (side opposite 30°) = k

= `6sqrt(3)` cm

2. In right ΔBCD (∠C = 90°, ∠B = 30°):

CD is opposite 30°.

Here, k = CD = 18.

So, BC (side opposite 60°) = `ksqrt(3)`

= `18sqrt(3)` cm

3. AC = AB + BC

= `6sqrt(3) + 18sqrt(3)` 

= `24sqrt(3)` cm

= 41.57 cm