Determine length from sketch

Question

I have a simple problem that I need to solve. Given a height (in blue), and an angle (eg: 60-degrees), I need to determine the length of the line in red, based on where the green line ends. The green line comes from the top of the blue line and is always 90-degrees.

The height of the blue line is variable. The angle of the blue line is variable.

Also, I do not know the length of the green dashed-line. Is there a way to figure out the length of the red line without knowing the length of the green?

-Adam

Any help would be much appreciated!

-Adam

Answer 1

Since you're dealing with right triangle, you can just use cosine function:

$$\cos(\theta)=\frac{blue}{red}$$

Or, substituting $60^o$ for $\theta$ and 10 for blue:

$$\cos(60^o)=\frac{10}{red}$$

Thus, solving for the length of the red side:

$$red=\frac{blue}{\cos(\theta)}$$

Or

$$red=\frac{10}{\cos(60^o)} = 20$$

Answer 2

Use Law of sines, $$\frac{a}{\sin{A}}=\frac{b}{\sin{B}}=\frac{c}{\sin{C}}$$

Let,

$\angle{A}=60^o$

$\angle{B}=90^o$

$\angle{C}=30^o$

$a=?$

$b=?$

$c=10cm$

From Law of sines we have,

$$\frac{a}{\sin{A}}=\frac{c}{\sin{C}}$$

Put the values and find $a$.

Now, from Law of sines we have,

$$\frac{a}{\sin{A}}=\frac{b}{\sin{B}}$$

Put the known values and find $b$.

Congratulations, you are done!

Answer 3

Reflect at the green line to obtain a triangle, which must be equilateral.