# Thread: Determining an exact expression for a length.

1. ## Determining an exact expression for a length.

A truss is used to hold up a heavy sign on a building, as shown.

The truss is attached to the building along side AB, which measures 2.4m. Determine an exact expression for the length CD.

My progress:

I found out that side a is approximately 1m. What the heck is going on with that expression though?

2. From your trigonometric functions, you can say that :

$\displaystyle tan(45) = \frac{AB}{BD}$, so $\displaystyle BD = \frac{AB}{tan(45)} = \frac{2.4}{1} = 2.4$

And also, of course :

$\displaystyle tan(60) = \frac{AB}{BC}$, so $\displaystyle BC = \frac{AB}{tan(60)} = \frac{2.4}{\sqrt{3}}$

Now, you know that $\displaystyle CD = BD - BC$, so $\displaystyle CD = 2.4 - \frac{2.4}{\sqrt{3}}$ ($\displaystyle \approx 1.014$)

$\displaystyle \rightarrow$ How did you find your result ? It might be interesting to see your reasoning

3. edit: Bacterius beat me to it.. oh well

Well I got my exact value and the answer is same as the one you provided. Here's how I did it:

To find CD you need to find BC and BD. Then subtract BC from BD.

You don't even need a calculator for this..

Set up the BD triangle and ignore the diagonal line through the middle.

The angle is 45 and we need to find the adjacent.

tan ratio:

$\displaystyle \tan 45 = \frac {2.4}{x}$

In this equation $\displaystyle x = BD$

$\displaystyle x = \frac {2.4}{\tan 45}$

We know that tan45 is 1 so...

$\displaystyle x = 2.4$

Now do the BC triangle..

The angle is 60 and you need to find adjacent.

Use tan again:

$\displaystyle \tan 60 = \frac {2.4}{x}$

$\displaystyle x = \frac {2.4}{\tan 60}$

We should know that tan60 is $\displaystyle \sqrt 3$

So $\displaystyle x = \frac {2.4}{\sqrt 3}$

So just put it together...

$\displaystyle 2.4 - (\frac {2.4}{\sqrt 3}) = CD$

that's how i did it?

4. Originally Posted by jgv115
edit: Bacterius beat me to it.. oh well
Sorry
I forgot to mention that triangle ADB is rectangle in B (it is written on the picture but it is better to mention it).

5. Thank you both. Btw I got 1 by first finding the length of AC then using that to find the length of DC using sin law. I've got what I need this question is so easy.. thanks

6. Originally Posted by kmjt
Thank you both. Btw I got 1 by first finding the length of AC then using that to find the length of DC using sin law. I've got what I need this question is so easy.. thanks
Yeah that's a way too ..