SOLUTION: A triangle ABC has sides AB=90 cm and AC=120 cm. A line segment DE is formed such that D lies on side AB while E lies on side AC. If the ratio of the areas of two triangles is 3:5,

Algebra ->  Triangles -> SOLUTION: A triangle ABC has sides AB=90 cm and AC=120 cm. A line segment DE is formed such that D lies on side AB while E lies on side AC. If the ratio of the areas of two triangles is 3:5,      Log On


   

Question 1151596: A triangle ABC has sides AB=90 cm and AC=120 cm. A line segment DE is formed such that D lies on side AB while E lies on side AC. If the ratio of the areas of two triangles is 3:5, how far is E from C if AD measures 60 cm?
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13209) About Me  (Show Source):
You can put this solution on YOUR website!


The statement of the problem is deficient. The distance from E to C can't be determined with the given information.

We can use AD as the base of triangle ADE and AB as the base of triangle ABC. Then we know the ratio of the bases of the two triangles is 2:3.

We can then determine the ratio of the altitudes of the two triangles, knowing that the ratio of the areas is 3:5.

%282%2F3%29%28a%2Fb%29+=+3%2F5
a%2Fb+=+9%2F10

So the altitude of triangle ADE has to be 9/10 of the altitude of triangle ABC for the ratio of the areas to be 3:5.

But there is no information given that allows us to convert that result into a distance between E and C.

Answer by ikleyn(52884) About Me  (Show Source):
You can put this solution on YOUR website!
.

Start from the formula for the triangle area


      A = %281%2F2%29%2Aa%2Ab%2Asin%28gamma%29,


where "a" and "b" are the lengths of any two sides of a triangle and gamma is the concluded angle between these sides.


For triangle ABC you have then  S%5BABC%5D = %281%2F2%29%2A90%2A120%2Asin%28A%29.


For triangle ADE you have   S%5BADE%5D = %281%2F2%29%2A60%2Aabs%28AE%29%2Asin%28A%29.



Now make the ratio of the triangles areas. You will get this equation 


    S%5BADE%5D%2FS%5BABC%5D = %2860%2Aabs%28AE%29%29%2F%2890%2A120%29 = 3%2F5.


From this equation (this proportion)


    | AE | = %2890%2A120%2A3%29%2F%2860%2A5%29 = 108.


So, now you know the length of the segment AE: it is 108 centimeters.


It implies that the length of EC is 120 - 108 = 12 centimeters.


ANSWER.  The distance from E to C is 12 centimeters.

Solved.