Find Length of line which has rotating object.

Question

I have 3 Images. A, B and C. if I place it on graph, its look something like this. Now main image is A and I place B and C on that image's (A) center point.

For easy understanding, let's consider only 2 images, A and B. Now I rotate image B to 10 degree as per below image. so how can I found value of line (Red or Yellow) ? (in first image, its 50, 50)

Answer 1

You have $\cos 10 = \dfrac{50}{\text{yellow}}$, so $\text{yellow}=\dfrac{50}{\cos 10} \approx 50.77$.

$\text{red} = 100 - \text{yellow} \approx 49.23$.

Your actual picture is rotated more than 10 degrees.

Answer 2

Hint:

let $A,B$ the two vertices of the left side of the starting internal square and $M$ the midpoint of $AB$ and $O$ the center. Rotating by an angle $\theta$ $A,b$ becomes $A',B'$ and $M$ becomes $M'$, $O$ is the fixed point. So $OM'=OM$ ( $50$ in your figure).

You want to find $ON$ where $N$ is the point of intersection of $A'B'$ with the horizontal axis.

Note that the $ON$ is the hypotenuse of the rectangle triangle $NM'O$ so: $$ \overline{ON}^2=\overline{NM'}^2+\overline{OM'}^2 $$ and $$ \overline{NM'}=\overline{OM'} \tan \theta $$ So you can find $\overline{ON}$.