SOLUTION: In the figure to the bottom, DE divides the area of triangle ABC in half. Find the length of EC in m. Diagram: https://imgur.com/a/dIZiJ6V

Algebra ->  Length-and-distance -> SOLUTION: In the figure to the bottom, DE divides the area of triangle ABC in half. Find the length of EC in m. Diagram: https://imgur.com/a/dIZiJ6V      Log On


   

Question 1151146: In the figure to the bottom, DE divides the area of triangle ABC in half. Find the length of EC in m.
Diagram: https://imgur.com/a/dIZiJ6V
Answer by ikleyn(52776) About Me  (Show Source):
You can put this solution on YOUR website!
.

Triangle ABC is a right-angled triangle.


Its leg AB is  sqrt%2825%5E2-7%5E2%29 = 24 meters long.


Its area is  %281%2F2%29%2A7%2A24 = 7*12 = 84 square meters.


Triangles ABC and EDC are similar (since they both are right-angled and have common acute angle C).


Hence, their corresponding sides are proportional.


In particular,  EC = k*AC = 24k,  ED = k*AB = 7k,  where "k" is the proportionality coefficient.


Then the area of the triangle EDC is  %281%2F2%29%2Aabs%28EC%29%2Aabs%28ED%29 = %281%2F2%29%2A24k%2A7k = 84k%5E2.


It is half the area of the triangle ABC, i.e.

     84k%5E2 = 84%2F2 = 42,   or


     k%5E2 = 42%2F84 = 1%2F2.


Therefore, the proportionality coefficient (the similarity coefficient)

     k = sqrt%281%2F2%29 = sqrt%282%29%2F2.


It implies  EC = k*AC = %28sqrt%282%29%2F2%29%2A24 = 12%2Asqrt%282%29 = 16.971 (approximately).    ANSWER

Solved.