Triangle ABC is an equilateral triangle. D and E are points on AB and AC respectively such that DE is parallel to BC and is equal to half the length of BC. If AD + CE + BC = 30 cm, then find the perimeter (in cm) of the quadrilateral BCED.

Triangle ABC is an equilateral triangle.

Let the length of BC = 2p

BC = AB = AC = 2p

DE is equal to half the length of BC.

Triangle ABC and triangle ADE are similar triangles.

$$\Rightarrow$$  $$\frac{AD}{AB}=\frac{DE}{BC}$$

$$\Rightarrow$$  $$\frac{AD}{AB}=\frac{p}{2p}$$

$$\Rightarrow$$  $$AD=\frac{1}{2}AB$$

$$\Rightarrow$$  $$AD=\frac{1}{2}\times2p$$

$$\Rightarrow$$  AD = p

Similarly, AE = p

and EC = AC - AE = 2p - p = p

AD + CE + BC = 30 cm

p + p + 2p = 30

4p = 30

p = $$\frac{15}{2}$$ cm

Perimeter of the quadrilateral BCED = BD + DE + CE + BC

= p + p + p + 2p

= 5p

= $$5\times\frac{15}{2}$$

= 37.5 cm

Hence, the correct answer is Option A