similar triangles/trig height question

A wooden ramp is being built to provide wheel chair access to the park. Stacey, Elias and Keiko drew out an initial plan where the ramp would span a horizontal distance of 9 m and a vertical distance of 2.7 m. However, they decided the ramp would need extra support. As a result, they decided to place an extra vertical support 5.5 m from the point where the ramp meets the ground

1. Use similar triangles to determine the length of the vertical support.
2. Use trigonometry to determine the length of the vertical support.

A picture is always helpful:

http://img359.imageshack.us/img359/2530/simtriby2.png

Make a proportion with the correlating sides.

That should be: $\displaystyle \frac{x}{5.5} = \frac{2.7}{9}$

Solve for $\displaystyle x$.

As for the trigonometry part, that's kinda tough (unless I'm just missing something big).

I would use pythagorean theorem to find the missing side on the big triangle,

then use law of cosines to find the angle the ramp makes with the ground.

Then use Cosine Law in the smaller triangle.

There must be a simpler way, but I don't see it.

Thanks
1.6 meters is the height?
For the trigonometry how can you use the cosine law inside the smaller triangle if you only have the base 5.5 and the height 1.6?

Quote:

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Originally Posted by Jonboy

A picture is always helpful:

http://img359.imageshack.us/img359/2530/simtriby2.png

Make a proportion with the correlating sides.

That should be: $\displaystyle \frac{x}{5.5} = \frac{2.7}{9}$

Solve for $\displaystyle x$.

As for the trigonometry part, that's kinda tough (unless I'm just missing something big).

I would use pythagorean theorem to find the missing side on the big triangle,

then use law of cosines to find the angle the ramp makes with the ground.

Then use Cosine Law in the smaller triangle.

There must be a simpler way, but I don't see it.

---

For the trig part, use the tangent value of the angle made by the ramp with the horizontal ground.

For the whole ramp,
tan(theta) = (2.7)/9

For the vertical support,
tan(theta) = x/(5.5)

So, since tan(theta) = tan(theta), then,
(2.7)/9 = x/(5.5) -----------**

It is the same as the by proportion part actually.

Wow, that's so much easier! Thanks Ticbol. I knew there had to something simpler.