How Can Similar Triangles Simplify Solving for x and y?

[triden]

Homework Statement

I've attached a diagram of the problem.

Homework Equations

Trig functions (soh cah toa) and pythagorus theorem a^2 + b^2 = c^2

The Attempt at a Solution

I've tried a bunch of things but I can't find a start. I know I can't assume that triangle ABC is a right triangle. I tried breaking it up into little triangles, but can't seem to find a good starting point.

 

[triden]

After looking at it, I guess I did figure it out, but not sure if it's a good way of doing it. I drew a line from A down to X. This formed a new triangle A,B,AX which is a 3,4,5 triangle. What if I didn't know that a right triangle with a "5" for a hypotenuse is a 3,4,5...do I have to have that knowledge to solve it? Thanks

 

[HallsofIvy]

Are we to assume that triangle ABC is a right triangle? If not I the problem isn't well-defined without knowing at least one angle.

I assume ABC is a right triangle then so is triangle ADE and we can use the Pythagorean theorem to find the length of AE. y is just 7 minus that.

Of course, x/5= 3.5/2 so x is easy to find.

What if I didn't know that a right triangle with a "5" for a hypotenuse is a 3,4

The problem is you don't know that. It isn't true and in this case the (right) triangle ABX you form is NOT a 3,4,5 right triangle. The triangle ABX is similar to ABC and its sides are not multiples of 3, 4, 5. There exist an infinite number of different right triangles with hypotenuse of length 5.

 

[triden]

Are we to assume that triangle ABC is a right triangle? If not I the problem isn't well-defined without knowing at least one angle.

I assume ABC is a right triangle then so is triangle ADE and we can use the Pythagorean theorem to find the length of AE. y is just 7 minus that.

Of course, x/5= 3.5/2 so x is easy to find.


The problem is you don't know that. It isn't true and in this case the (right) triangle ABX you form is NOT a 3,4,5 right triangle. The triangle ABX is similar to ABC and its sides are not multiples of 3, 4, 5. There exist an infinite number of different right triangles with hypotenuse of length 5.

I don't think you can assume that it's right because if you go 2^2 + 7^2 = c^2, you end up with 8.6 which is the wrong answer. Your method for finding X works...just using the equivalence formula. It gets 8.75 which is correct.

 

[Chaos2009]

Um, you guys are missing something pretty big. You don't need any trig functions or pythagoreah theorem. We are told that DE and BC are parallel. From that, we can prove that the two triangles, ADE and ABC, are similar. We can use the properties of similar triangles to solve for x and y easily. I hope that helps.