The base of triangle ABC is one half the altitude. The altitude of triangle DEF is one half its base. If both triangles are equal in area, what is the ratio of the altitude of triangle ABC to the altitude of triangle DEF?

Question

a. 1:4
b. 1:2
c. 1:1
d. 2:1
e. 4:1

please answer and explain

Answer 1

If we call the respective bases and altitudes a,b for ABC and d,e for DEF, then we have

a = b/2
e = d/2, so d = 2e

So, let's find the ratio b/e
Since the areas are equal,

(1/2)(ab) = (1/2)(de)
(b/2)(b) = (2e)(e)
b^2/2 = 2e^2
b^2/e^2 = 4
b/e = 2

So, (D) 2:1