Area of triangle

[thales123]

How do I find the area of the black triangle below? The area of the square is 4k.


And we also know that the lengths of the legs of the gray triangle below are 2√k and √k.

 

[Dr.Peterson]

Look for a triangle similar to the black one.

 

[thales123]

Look for a triangle similar to the black one.

There is the triangle ABE, whose area is 4k. Since those triangles are similar, we can say that AB/CD=4, which means that the ratio of their areas is 16. But I don't know what else to do!

 

[Dr.Peterson]

Aren't you finished? If ABE has area 4k, and ABE:BCD = 16:1, then what is area BCD?

On the other hand, check whether your area for ABE is correct. I believe that the area of ABC is 4k.

You could also have used the similar triangle I had in mind, which is BDC (or the other unnamed large gray triangle that is congruent to it).

 

[thales123]

On the other hand, check whether your area for ABE is correct. I believe that the area of ABC is 4k.

My bad. That's what I meant to say. The area of ABC, not ABE, is 4k. But the black triangle is similar to ABE, whose area we don't know.

 

[Dr.Peterson]

Look again. ABE, ACB, BCE, BDC, CDE, and others are all similar!

 

[thales123]

Look again. ABE, ACB, BCE, BDC, CDE, and others are all similar!

But how do I prove that CDE is similar to BCE?

 

[Dr.Peterson]

Look at the angles. (I wouldn't use BCE, but the same is true of whatever triangle you use.)

 

[thales123]

One way to solve the problem would be to find information on the area of BCE. The original source states that EB/ED=4, but I'm not so sure about that.